A decade ago I drove the area in a cutu-ute (Element).
In Moab I drove down Long Canyon without problem (low range would have been nice, but not essential)
In Teluride I drove a short ways UP Black Bear, just enough to get a good view of slope. Since there was a festival in town (Blues and Brews), the campground in town was full. Instead I drove up to Alta Lakes (typical Forest Service road) where we spent a blustery night.
I was planning on driving some of the easier passes (possibly even to the top of Black Bear as proposed by the OP). The Element doesn't have much clearance, but has a great turning radius - so turning tail when the going gets too rough is an option). But with the storm, I decided it was wiser to stick with paved roads, and instead looped around to Ouray and Silverton and down to Durango and Cortez.
On a previous trip I'd taken a S10 (extended cab, 4x4) from Silverton to Ouray via Animas Forks. I was intending to take Engineer Pass but there was snow on the north side.