It's sort of a backdoor, because while you would be living in Mexico in the sense that Mexico is where your apartment is, that apartment could be less then a mile from a border crossing so you could easily do most things in the US and just go back to Mexico every night to sleep.

If the internet is to be believed they would not need a Temporary Import Permit when visiting the US with the Mexican car because they are a US citizen. They would be able to stay with the car for up to a year in the US, so even if they wanted to take a vacation or road trip to someplace other than Laredo, TX that should be OK.

PS: there's an interesting limitation of Google maps I just noticed. If I've got a view that shows both the US and Mexico Laredos and do a search, I only get results on the US side. I have to scroll so that most of the view is the Mexico side before it will show anything.