You could google the Wind River Indian Reservation and find out how much their fishing permit (required for reservation access whether or not you fish) currently is per day. I've read (in Nancy Pallister's "Beyond Trails in the Wind River Mountains," the only guidebook I've seen that addresses the recent reservation issues) that you need two one-day permits, one for the day you go in and another for the day you go out.
The Big Sandy trailhead is a lot closer to where you want to go, and there are no permit problems! It's also closer to Utah (less driving time). What you'll miss by going from Big Sandy is the high point of the Bears Ears trail. You could skip Texas Pass and take a longer trail to the Cirque of the Towers via the Lizard Head Plateau instead, which would make up for missing the Bears Ears Trail. I'm hoping to make that loop this summer.