From what I can gather, the Texas Motorist Protection Act, HB 1815, allows for a person (with or without a CHL), to carry a handgun in their car as long as:
1. the handgun remains concealed
2. the person is not engaged in criminal activity, other than a Class C misdemeanor (a violation of a law or ordinance regulating traffic)
3. the person is not prohibited by law from possessing a firearm
4. the person is not a member of a criminal street gang
Unless you have a CHL, you cannot carry the weapon outside your vehicle unless you are going directly to or from your car and home or a range. IE, if you stop off at Walmart, leave it in the car unless you have a CHL.
I'm no lawyer, haven't played one on TV, nor did not sleep at a Holiday Inn Express last night, but your idea seems to be outside the "handgun remains concealed" requirement.
You might want to check with someone in law enforcement to be sure, but I'd say it would be illegal.