Yes, there are muddy parts, especially in the rain. Some of the path is dirt with ruts. You're bike may work, but maybe with wider tires if you want to do that. But I think you could take the Mongoose (provided you can carry your stuff on it). It's not all that tough of a ride and if you're going with the Boy Scouts, I doubt you'll be moving very fast. I'm sure many of them will be on Mongooses (or is it Mongeese?). My friend (300+ lbs) did in on a 20-year-old Huffy in 2.5 days, smoking and drinking Mountain Dew the whole way.

When I did the trail, it rained on the first day (from Cumberland) and there was some mud but it wasn't bad. It gets a little worse closer to DC but I don't think it would be unmanageable at a Boy Scout pace with skinny tires.