Local governments can ban bicycles from certain roads under their jurisdiction, but certain things must happen before these bans become effective -- in particular, there needs to be signs, and for minimum speed regulations (which could prevent the use of a road by a bicycle if they were determined to apply to bicycles too -- which some court cases in other places have said they do not, but I'm not sure if anything applies here) there needs to also be a study. Relevant laws could be --

284.071. CONTROLLED ACCESS TO TOLL ROAD.
545.065. STATE AND LOCAL REGULATION OF LIMITED-ACCESS OR CONTROLLED-ACCESS HIGHWAYS
545.0651. RESTRICTION ON USE OF HIGHWAY
545.363. MINIMUM SPEED REGULATIONS.

I don't live near Houston and am not at all familiar with the local laws there, but my guess is that either 1) there is no such restriction and the officer is flat out wrong, or 2) if there is a restriction, it doesn't jive with Texas law. (If there's no signs, it can't jive with Texas law.)

My advice to you would be to contact a local bicycling advocacy group and see what they say. Here in Austin I'd say the LOBV would be a good place to start -- maybe Houston has something similar?