Can't say about your TC specifically or how brave you are. When I drove the Needles, people were driving their motorhomes and truck campers through the tunnels, despite warning signs not to. There's more than one video on youtube about RV'ers in trouble on the Needles.
I have a map of the road, with the ht / wt of each tunnel listed, but I'm not finding it online...
strike the above, here's the map:
http://gfp.sd.gov/state-parks/directory/custer/docs/custer-map.pdfEdit - Looking at the post below, looks like I made the same mistake, or rather, lumped the Needles and Iron Mtn. together (should drive both if you can!)