If you live in LA, then you owe LA state income tax on all your world-wide income. It sounds like you are saying that your TX-based employer does not want to go to the trouble of withholding LA income tax from your paycheck (which would also require sending various forms and funds to LA).
I can't say whether or not LA would object to the employer's refusal, but that is their problem. You should be able to avoid penalties in LA by making estimated tax payments at least quarterly, as described here. https://revenue.louisiana.gov/IndividualIncomeTax#det
Hadn’t thought about doing quarterly estimated taxes just for LA but that makes sense. I appreciate the lead to check into!
If you have Louisiana income and don't have withholding, you are required to make estimated payments, or else you will be assessed a penalty, even if you pay in full when you file your tax return in April.