this is what i would do. If i had to stand by and watch this happen i would eventually take drastic action to prevent it, and i don’t mean taking him out of the country.
THe problem is, if you are leaving the country they will stop you at the airport with your kids unless you have written consent from their other parent to leave. I saw this happen with a guy I worked with. His wife was making their yearly pilgrimage to Greece and he was going to catch up to her in a couple weeks. THey were both archaeologists and had been going to Greece to dig every summer for almost a decade. This time he forgot to do the form and had to scramble to get it signed, notarized, and to the airport before their flight left.
It’s pretty easy to get out of the US via ground travel.
But honestly, the answer is not to leave the country. It is to remove the mother’s authority - by whatever means necessary. The courts, of course, would be the first resort. That’s “plan A”.
And he did plan A and it failed. He may have to move on to plan B, and maybe even plan C.
Texas is close to Mexico. Cross the border and fly out from there?