I know, that's what concerns me. Neither of us have definitive legal rights as of today...
Plus if we do it now, she can't afford a very good lawyer. I on the other hand, can pay for one right now. So the question is be preemptive and potentially have the upper hand, or wait.
You are all over the place with notions of filing preemptive legal attacks, and what you feel you will accomplish with this gambit.
You were never married to the mother, and there are no custody orders in place.
As of now, The Mother Is The Only One with a legal possessory and custodial right to your child.
One pretty sure bet is that very soon after she is served, you will cease to have your child as much as you now do. Certainly far less than the half time you now claim. This too would be a preemptive move that her attorney would probably encourage, in anticipation of the pending hearing.
It is very unlikely a Texas judge will grant 50/50 custody in a situation where the parents are antagonistic, and much less to those who were not married. The most probable result of your going back to court to establish custody orders would be that mom would be the custodial parent, and you given a Texas Standard Possession order and child support order.
While you certainly can and should go back to court to amicably get a custody order in place, any ill advised attempt to blindside her and legally force 50/50 would almost surely be a bad idea.