Unusual military bike
OK, you hit my soft spot...military bikes. The most recognizable is the WWII Harley WLA. If you can scare up a frame and a fork, (The most expensive parts) there are a lot of surplus engines out there still, as well as plenty of parts. Anything scarce is now reproduced. They are easy to work on, and a blast to ride (But you have to get used to the foot clutch and tank shift). They ARE NOT suicide shift...that is a whole different setup from the chopper days. There is a great WLA forum out there. In the end, you will have a universally recognized and respected bike worth good money.
Military Russian Dnepr or Ural sidecar motorcycle rig (Preferably the flathead 750cc model) is a really fun bike, and affordable still. BEWARE the Dnepr kit bikes from Ohio, they are NOT DOT legal, and can be a big hassle to register. The flathead models (Pre-1968 or 1971...I forget the DOT cutoff) are easily registered. They usually need a fair amount of tikering to get them reliable, but once they are straightened out, they are reliable. Before the fall of the Iron Curtain, the bikes were well built. Afterwards they were junk. Early overhead valve Dneprs (Pre- Iron Curtain falling) were also good machines. You can load them up with dummy machine gun, ammo can racks, etc. Plenty of cheap parts available, and excellent enthusiast forums to support you.
Ducati Condor from around 1973-1978 or so. Old school technology, not really expensive, and fun to ride. A rare bird, and parts can be pricey.
The Rotax powered Harley of the early 1990's is a rugged, usable bike that just seems to keep going up in value. Doesn't have the vintage mojo.
The good old BSA M20 is a slow, simple but FUN militray bike to ride. Parts are still available, and ratty incomplete ones can still be bought for cheap money. When complete and running good, they are bringing some decent money today. they used to be cheap-cheap-cheap but are moving up.
OK, here's the ticket. In the late fifties and early sixties, Harley made some military issue Sportsters called the XLA. Rare as hens teeth. You could buy a cheap Sportster and make your own replica XLA. So few people have actually seen one in person, so most won't even be able to pick it apart.
I have a friend with a 1967 Harley1200 police bike from Kuwait, complete with bullet hole.
Sounds like a fun project you are embarking on!