Santa Fe to Taos to Great Sand Dunes NP

OK and the pad length is 35 feet.

Campground is in grove of Pinion pines.
Pinion Campground

How many routes are there? 160 in CO is the only way unless you are going to take the 4WD road from the north side of the park. And 285 out of NM would be the normal way unless you are going to take local roads. Or backtrack to the interstate.

As for the Gator farm, coming from Florida the idea of a gator farm in high desert Colorado is bizarre. Like finding a Polar bear farm outside the town I live in. We've been by it several time but never tempted to stop.

It's been awhile since we camped in GS but I thought it was open sand fields.

The campground in Great Sand Dunes could be tight for your rig. May want to drop toad and drive it through.
Visit the alligator farm just north of park. Interesting, but kind of tired. Gator ranch