I have done some riding in the Big Bend area in the past and there are a few options for you if you want to get to the Big Bend Expedition a day ahead or stay after our ride.
Lajitas Resort - The closest to the ride start and is a very nice resort with classic Texas style old west rooms and a beautiful golf course right on the Mexican border. They have a cafe onsite as well some some shopping. https://www.lajitasgolfresort.com/
If you stay extra, this area of Texas offers some really nice road cycling opportunities with some epic climbs and maybe the toughest climb in Texas West of Lajitas toward Presidio.
The old ghost town of Terlingua offers a few lodging opportunities with a few small hotels and some bed and breakfast style lodging and camping opportunities. It is also home to the Starlight Theater and an opportunity to see the most unique town in Texas perhaps. https://visitbigbend.com/terlingua/