I'm not sure if Texas Parks & Wildlife has discussed bounties to encourage culling. Such a program would probably require increased taxes to fund, which is not likely to be popular... though with the apparent increasing coverage of the issue, maybe that wouldn't be a problem. If everyone is on board, then it's just a question of how much the people (read: the government) are willing to spend.
I would also be concerned about unintended consequences, like you mentioned. I know for a fact that there are people today who, if they manage to trap the right pigs, would try to sell them. Their market would be landowners that offer canned hunts on a high-fenced property (say, $300 for a guaranteed large hog kill). Sure, the hog is still dead - but what if it breeds first? What if the piglets then escaped? Personally I'd feel dirty if I didn't cull the animals I trapped immediately, but if I were less financially secure, maybe not.