OK! I checked your tow vehicle and trailer specs, and you can probably make the US 64 climb to Taos. There don't seem to be any length restrictions on this section of the road. The Eagle Nest side is steep but the Taos side is relatively gentle. There are several RV Parks to stay at in the region. We've stayed at the Golden Eagle in Eagle Nest and that is a good jumping off point for exploring the area. The couple who own the place are really nice folks and run a nice place. We will be going there in September for a few days. There are a couple of RV parks down at Angel Fire and a couple along US 64 between Palo Flechado Pass and Tao. I would avoid the Plaza area of Taos with your rig. Taos is VERY congested. There are RV parks there but you would need to take the bypass. The streets are narrow making steering your rig a bit daunting.
If you decide to go by the Red River route, remember that Taos traffic is VERY heavy during the day and will be a bit slow! Enjoy your trip and contact me if you have any more questions. We've spent a lot of time there and my wife has family in both Taos and Cimarron.