I would check with your airline first, but we have had guys do what you mentioned.
Also had guys put it into a soft-sided cooler and carry it on.
Most airlines don't want any liquids dripping out of your cooler, so either way you do it, I would try to get the meat into ice and keep draining the water/blood till the meat is washed clean, if possible.
Get it chilled or partially frozen if possible, and triple bag it to prevent leaks.
You can add newspaper over the top of the bags to fill the cooler, and help insulate it.
I have brought home Lobster from Belize, Caribou from Northern Quebec, Fish from Canada.
The airlines may want to look, so carry some duct tape with you.
Your TX outfitter may have some ideas too.