Yes, technically that spare AVL engine will fit any Redditch heavyweight frame from a 1949 350 Bullet to a 1970 Series 2 Interceptor.
I see that 350 at Hitchcocks' doesn't have a V5C, and a check on gov.uk shows it's not on the DVLA computer. With some help from the owner's club you might be able to get an age-related registration number, or even POSSIBLY get the old number back, though the illegible frame number (stamped on the LEFT HAND side of the headstock, very careful cleaning of that area might bring it back) won't help. The club will want to see evidence of an original bike ready for a MOT, so that will mean keeping the original engine in place and getting it going. Once it's an MOT'd and registered runner, then you can change the registration status to historic vehicle and enjoy tax and (further) MOT exemption, and THEN you can swap engines!
There is always the risk that if you got that one running you might enjoy it as it is. An apparently decent registered and running 1961 350 Bullet sold for £1100 in South Wales last Thursday, I did post a link to the eBay listing in the vintage section a few days earlier...
I didn't see any RE frames with or without log books at Netley Marsh or Somerset VMCC autojumbles this year, but next year? Also if you join the RE Owner's club your get to see the small ads in the magazine as well as what's on the web site.
A.