I don't know about Tx Children's and I don't know what your specific needs are, but one of my sons has had eye issues since he was born and Dr. Miller at Houston Eye Associates has operated on him 3 times since he was born, and I can't say enough good things about him. My son is 12 now and thriving.
Not sure if they still have it but Baylor college of Medicine had an eye center in Houston when I was a kid and I had a lot of weird eye issues. That was my go to.
TCH doctors are BCM doctors. Their actual employer and professorship is through BCM, even if they only work at TCH proper.
I think a fair number of the ophthalmologists and oculoplastic surgeons see children on the TCH side and adults on the BCM side. I know that the doctor I see - Richard Allen, who I definitely recommend - does that.