This is likely institution/program dependent rather than state dependent. After a quick perusal, TX doesn't have a rule (like CA does) that anyone over PGY2 level must have a full license. Your program or GME office however may have more restrictive requirements. Check with them.
That said, I would just go for it personally. You are going to have massive amounts of time 2nd and 3rd year of fellowship to moonlight and you can't do that without a license. Also, I doubt TX is more of a hassle to get a license in than any other state so eventually you're going to have to nut up and deal with what is, just about everywhere, an enormous hassle.