The smart money is to get a Pre-Unit Bullet and bolt on a Hitchcocks Hardtail kit. That way you'd end up with something rideable and still have a farm left to mortgage later.
https://accessories.hitchcocksmotorcycles.com/19209?ref_page=Bullet%20Export%20500ccPART No. 99150 ; RIGID REAR FRAME WITH FIXINGS ; £540.00
A substantial rear section specifically developed for the Indian Bullet frame made from cold drawn seamless tubing. This is complete with the top fixing clamp arrangement and full instructions. Supplied un-painted. Made in the UK.With 1/2 width hubs and a nice Indian-sourced fuel tank, It'd really look the part. The motor would be able to keep up with modern traffic.
Blue Moon Cycles, Norcross, Georgia usually has a nice selection. The substantial prices exist because it takes a LOT of dinero to make them look like they do when starting with a barn find.
https://www.bluemooncycle.com/motorcycles-for-sale/The reason that the Bullet is an affordable, rideable machine is because Allan Hitchcock created an accessible parts supply. Otherwise you are playing the NOS game looking for non-fermented hardware remnants made in 1937. If the idea is to acquire a machinist friend and put his kid thru college, restoring a barn-find is a good start. The geniuses like Grumbern are busy building their own ideas from scratch, and they are a rare commodity anyway.
Taking your hand-built, fully restored BSA 350 sidevalve out into traffic isn't much fun when you twig to the fact that one trip into a ditch takes about $10K right off the top. If you are dealing with Jay Leno resources & money, none of this an issue. For the ordinary guy the Pre-Unit Bullet is the most affordable antique you can own and actually use.
If you are going to restore & not use, it's way cheaper to just go to the Antique Motorcycle shows and make some wall-sized pictures of your favorites. You can dress up a rigid frame equipped Bullet any way you like, and 99% of the passers by won't know it's not a historical artifact, just like your trail bike does right now. If you'd started your trail bike project with a BSA 441 Victor or the like, things would have been a lot harder and more expensive to sort out.
Plan "B" might be to just acquire an old running complete machine from England. At least all the major bits are there and you don't have to start playing parts fabricator right away. Hitchcocks has a close-to-operational 250 Model S for about 2800 pounds, and a 350 Model G semi-intact parts bike for 1200 pounds. There's a LOT more work and parts that 1600 pounds probably wouldn't cover between them.
https://accessories.hitchcocksmotorcycles.com/Used-Parts-and-Bikes/Bikes/44026