I would call the rep back and ask him/her to give you the cite for the law or whom to contact at the TX State and ask directly. That doesn't make sense. Why would TX care where you register your car? If people buys a car in TX because they were in an accident, which totaled their car, are they saying that you can't buy the car in TX and take it home to register it? Makes no sense. Tesla should be able to get you a temp tag so you can drive back to Louisiana and then register it there.
Here is what I found:
Buying a car in another state involves paying the local Texas auto sales tax, the sales tax difference between Texas and your home state, and any processing fees charged by the DMV. Beyond that — and the price of the vehicle — you won't have to worry about any additional fees and taxes.
https://www.astonmartinhouston.com › car-buying-advice
Can I buy a car in Texas and register in Louisiana?
If you are a resident of Louisiana,
you are free to purchase a vehicle in another state, but it will incur fees and frustrations. According to the Office of Motor Vehicles, you must register your vehicle in Louisiana to legally drive on your home state's highways.Jul 30, 2019