Squeezebox, Generally wider is better, but not necessarily meaning a fat tired bike.
I had some 35 mm Panaracer Crosstowns on my touring bike and they were almost too wide to fit between the chainstays. They handled even ungroomed trails very well. On a different touring bike, but with the same frame, I have 37 mm Continentals that are much more narrow. Not quite as good on ungroomed trails nor would I expect them to do very well in loose sand. The bike that had the Crosstowns now has 32 mm Continentals and they're quite adequate for groomed trails.
A '90s era fully rigid mountain bike converted to a touring bike would be a very good choice regardless of the road surface you plan to tour on, even more so if you plan to ride the R2T routes.
Brad