After looking at a map of the area fairly closely, you don't need 4wd to get from Pitkin up to the tunnel, altho high ground clearance is nice to have. This trip is over FR839.
From the east side, it is an easy drive to St. Elmo. No problem at all for a car. FR295 from St. Elmo to Hancock is a "pain in the butt", in more ways than one. Very rough, ungraded, chug holes (bigger than a pot hole), etc. Definitely a high ground clearance and slow going road. After reaching Hancock, the old roadbed to teh tunnel has been blocked by the Forest Service so you have to walk it.
If you meant FR 299, FR266, and FR888 (as shown on the map I'm looking at) over Hancock Pass, then it is definitely a 4wd trip. This is one of the roughest passes that I've ever had the "pleasure" of traversing. It's best done in late summer as there is a boggy mess on the east side.
Hope this answers some of your questions. If not drop me a line.