I did find this note on the WYBDR site, that suggests the entire route *should* be doable with a truck or other high clearance 4x4:

Wyoming Backcountry Discovery Route (WYBDR) Map & Information (ridebdr.com)

All of the roads used on the WYBDR are open to vehicles. If you do decide to drive this route you will want a 4×4 with high clearance as the route has deep puddles, large rocks, huge ruts and steep loose grades. Always stay to the right and go slow around blind corners to avoid head-on accidents with others using the roads. We recommend traveling south to north to minimize head-on traffic.
When we filmed the WYBDR documentary our support vehicle remained on major roads as much as possible. 

It looks like there is a film screening coming up in March in Portland, or the online release coming on May 1. From the trailer, looks like a beautiful and amazing route.

Jason