Since I live about 170 miles north of the City of Manor, I can only fantasize about protesting in person their bicycle ban. But I have some ideas about how to make some noise about it if I lived in Austin.
I would drive my bicycle up to the edge where bicycle use is prohibited. I would dismount and take off my cleats and socks. I would then cross the road to walk to the end of manor’s jurisdiction facing traffic. I would use the roadway as a pedestrian for the length of the banned portion of the road, and then remount to continue on my way.
I would not walk on the shoulder, as that would be far too hazardous for me do in bare feet. (I define the word “tenderfoot”)
I would do it as often as I could. I would be especially keen to do this during rush hours. I would recruit my friends so we could do it in groups.
This would be a passive-aggressive gambit. I want to legally obstruct the traffic flow to the extent that the local travelers would rather I be riding my bicycle than walking it.
Because it is a public road, it is open to every form of traffic, except that of bicycles, which are banned. Pedestrians are traffic according to the Texas Transportation Code. (TTC) 
I would expect that the police would be called. If they did not show up by the time I traversed the banned section, I would call and complain about pedestrians on the road myself! (In an especially whiny tone of voice, by the way.)
If the good citizens of Manor have arranged to get bicycles off their precious road, they will no doubt feel wronged if a cyclist can walk upon it. Complaints to the local constable may ensue, and it will be good to establish the legality of pedestrian use of the road early with them.
On the road, as a lawful pedestrian, I would end any interaction with police as soon as possible. Outside of giving them my real name and address,  I would not give them any other information. I would ask a lot of questions, and write down the answers. I would write down their name. I would ask why they were stopping me. What law am I in violation of? Can I read the ordinance they are accusing me of breaking? Is our conversation being recorded? What is the procedure of obtaining a copy of this recording?
I would continually ask to be allowed to continue on my way. For example;
They might ask you where you are going. Answer: “That away.” Point up the road. “Can I be back on my way now?”
They might ask you where you are coming from. Answer: “From back there.” Point back down the road. “Can I be back on my way now?”
They may ask you what you are doing out there on this road. Answer: “Traveling on a public road. Can I be back on my way now?”
If they ask you why you didn’t use an alternative road, I would say something like “I chose to use this public road, as is my right, is it not? Are we done here?”
An alternative answer to any of those particular questions would be to answer with a question like this; “Did you stop me for that purpose? To ask where I work, where I am going, where I am coming from, to find out why I am here? May I continue on my way then, please?”
They may say that you are free to go, and then ask you if they could talk to you about it. Take the opportunity to leave. They simply want to tell you to stay out of the way, blah blah blah. A safety lecture. There is no upside for you, and you will be tempted to lower your guard and say things that are not in your best interest.
Provided they don’t make up a law, they will have no reason to give you a citation or a warning. They will settle for a stern safety lecture. I would promptly complain to the officer’s supervisor over this. Stopping a lawful pedestrian under the penalty of law (You have to stop or be subject to arrest.) to be given a safety lecture is an abuse of power.
If I lived a reasonable distance from Manor, (And I had an automobile) I would often drive through that section well below the speed limit. There is no minimum speed limit.  The Maximum Speed Limit law  should be employed to justify your slower speed through there.  Be familiar with the Slow Moving Vehicle laws of Texas and be in compliance with them.  Be sure your automobile is current with registration and insurance, and that all lights are in working order.
If you are subject to a traffic stop in your automobile, only answer questions that are pertinent to your stop. For example; “Why are you driving so slow?” (“They banned bicycles on this road due to its condition, so I am just being reasonable and prudent for a road that is in such bad shape.”)
“You have to keep up with other traffic!” (I am required by law to drive in a prudent manner and with due care. I am not responsible for the reckless behavior of other drivers. Why are you allowing them to operate in such cavalier manner at this time and place?)
“You are holding up traffic!” (I am traffic. I am traveling as fast as is safe for the current conditions and potential hazards. The other traffic is violating the maximum speed law of Texas. I would think that your beef would be with them!)
“You have to drive faster through here!” (Are you instructing me to operate my car at a faster speed than I am comfortable with?”) 
Not A Cause For Austin?
I hear Austin has a vibrant cycling community. Well connected and engaged in advocacy issues. Perhaps they are OK if a community near them bans bicycles from an otherwise public thoroughfare. They wouldn’t ride there anyway! After all, Manor has no three foot ordinance to protect cyclists, and there are no bike lanes that extend all the way to the City of Manor. It would be impossible to ride there on a bicycle!
What good is there in having a three-foot law when you are not allowed on the street in the first place? What’s the use of getting more butts on bikes when they can only ride to destinations connected with a MUP? Why is this not an issue with you? (Queue the sound of crickets)
Other Things To Do
The City of Manor has a website and a facebook page. Call, write and e-mail them asking when the “temporary” ban will be lifted. Ask them when the road repairs will be completed.
Request that the local news media follow the issue.
Show up at Manor town meetings and request time to petition the town officials about the issue.
Publish pictures of the signs and the “damaged and dangerous” road.
Cyclists and bicycle advocates of Austin; Don’t you think it is about time to make some noise over this? Or are you content to leave well enough alone, hoping they will ban bicycles on your street last?
 Sec. 541.301. TRAFFIC. In this subtitle "traffic" means pedestrians, ridden or herded animals, and conveyances, including vehicles and streetcars, singly or together while using a highway for the purposes of travel.
 In Texas, one must give to a police officer who requests it his real name and current address. You are not required to carry ID, and in this situation I would not. However, if you do not produce evidence that satisfies the officer that you are telling him who you are truthfully, he can detain you until your ID can be confirmed. When you are operating a motor vehicle on a public road, you must be able to produce a driver’s license at any time.
 Sec. 545.363 (c) If appropriate signs are erected giving notice of a minimum speed limit adopted under this section, an operator may not drive a vehicle more slowly than that limit except as necessary for safe operation or in compliance with law.
 Sec. 545.351. MAXIMUM SPEED REQUIREMENT. (a) An operator may not drive at a speed greater than is reasonable and prudent under the circumstances then existing. (b) An operator: (1) may not drive a vehicle at a speed greater than is reasonable and prudent under the conditions and having regard for actual and potential hazards then existing; and (2) shall control the speed of the vehicle as necessary to avoid colliding with another person or vehicle that is on or entering the highway in compliance with law and the duty of each person to use due care. (c) An operator shall, consistent with Subsections (a) and (b), drive at an appropriate reduced speed if: (1) the operator is approaching and crossing an intersection or railroad grade crossing; (2) the operator is approaching and going around a curve; (3) the operator is approaching a hill crest; (4) the operator is traveling on a narrow or winding roadway; and (5) a special hazard exists with regard to traffic, including pedestrians, or weather or highway conditions.
 As you can see in the footnote above, (b)(1) one cannot drive faster than prudent for potential hazards that may exist. One hazard present is road surface conditions so treacherous as to necessitate the banning of bicycles. It is reasonable to expect that motorists could encounter pedestrians on the road as well, a condition (c)(5) in the statute above, in light that cyclists cannot ride their bicycles there.
 Sec. 545.051 (b) An operator of a vehicle on a roadway moving more slowly than the normal speed of other vehicles at the time and place under the existing conditions shall drive in the right-hand lane available for vehicles, or as close as practicable to the right-hand curb or edge of the roadway, unless the operator is: (1) passing another vehicle; or (2) preparing for a left turn at an intersection or into a private road or driveway.
Note the word “or” in the above statute. If you are in a motor vehicle, and you are on a laned roadway, as the road in question is, you are in compliance with this statute if you are in the rightmost lane available. If the roadway is unlaned, then the “curb or edge of roadway” part applies.
Sec. 545.363. MINIMUM SPEED REGULATIONS. (a) An operator may not drive so slowly as to impede the normal and reasonable movement of traffic, except when reduced speed is necessary for safe operation or in compliance with law.
By driving slower than other traffic on the road, you are claiming that, in the first place, the road is too dangerous for bicycles according to the City of Manor and their posted highway signs. Prudence would dictate that automobile operators should use extra caution on a road whose surface is in such bad shape.
Secondly, it would not be unreasonable to expect to encounter a pedestrians on this road, as cyclists would be afoot and there is no sidewalk. A special hazard, a pedestrian, may exist.(See Sec.545.351 (c)(5))
 Traveling at the posted maximum speed limit is only allowed when conditions are perfect. Anything that would mar that perfection will demand a slower maximum speed limit than the posted speed limit. Perhaps the Officer could explain to a judge how a road who’s surface is too dangerous for bicycles to safely traverse is simultaneously the perfect road conditions for a car, thus allowing the posted maximum speed limit to prevail?