The route over Gunnison and down into Salida is over Monarch Pass. We towed our 32' TT with our F250 V10 6.8L and it was a challenge down the other side. We would not hesitate with our diesel now and our heavier FW because we have tow/haul mode, but with the gasser, we didn't. This was the most challenging for us, and we've been all over the places you mentioned towing with our gasser - Hwy 12 all from Capitol Reef to Bryce. No problems heading to Monument Valley and around that area.
Since you mentioned the Four Corners area and Salida - somewhere in between is Ouray. There is a steep and winding pass between Ouray and Silverton. We towed to Ouray, parked our rig at a KOA there, toured with just our TV up over the million dollar highway to Silverton, Four Corners, Mesa Verde, then back to Ouray. We returned the same route towing as we towed in, in order to avoid towing over that pass south of Ouray.