You may want to look at the CDL and the registration of the vehicle as two separate issues.
If registering the vehicle correctly (I prefer the Historical Vehicle approach) the CDL becomes a non-issue. Unless, of course, you want or need one anyway.
To register as an RV in CA can be tricky if the vehicle was originally made to pull trailers (tractor) or haul loads (truck). With enough amenities such as a toilet, shower, kitchen, etc. it's theroretically doable. Missing the latter two, and having a 5th wheel, I never managed. Come to think of it, I tried without the 5th wheel, too, and failed.
They may want you to run in an occasional parade with the Historical plate, but I doubt that even the NSA would know if you didn't.