Cheap as possible? - well, that depends on how good you are at scrounging. I have about 175 bucks in parts on my 29'er. New engine, new sprocket, new chain, new 203mm rotor..
Possible. But it will look like a frankenstein erector set job, and it will be a lot harder to build. Perhaps you could make friends with a neighbor? Or fab the parts & weld everything at the storage locker with the genset & welding machine?
Any FWD bike needs to be a solid build, because it can really hurt you if it fails. A life altering injury - check out what happened to Bairdco when he had a front end failure. It ended his career, put him out of commission for months: