Mid may will be just fine. Snow can happen during this period of time, but it's fairly unusual, and it's wet stuff that melts almost immediately.
Driving this route, which we do all the time (usually in April), has only two high country areas: Palo Flechado Pass between Angel Fire and Taos has some twisty road, but it's brief, only about two miles. The high country between Tres Piedras and Tierra Amarilla is not a pass per se, just a drive through some high-elevation territory. Both are beautiful, but the latter offers just spectacular views of the mountains from the very large pull-offs on the right-hand side of the highway.
Taos can be congested, not having been laid out hundreds of years ago for modern traffic. Here's my advice:
When you approach Taos from the east, take the by-bass. This will lead you AWAY from your destination, but right back to the main street through Taos. You turn right on the main street after the by-pass and drive straight through town. In this way, you avoid having to make any turns down at the really congested part of town at the plaza. All in all, it's a mile or so longer, but a major headache less.
After Chama, you'll pass through the northern part of the Jicarilla Apache rez and its most major town, Dulce. Very pretty country, too!
🙂Lynn