Main problem would be the price, but it sounds like you're looking at that correctly.
Electrical issues can run the gauntlet. Simple or nightmare. This one is a relatively minor subsystem, sounds like it's straightforward and should be easy to isolate, but I've heard that before. There's an electrical diagram in the owner's manual for the Hawk that may help to get to the bottom of it, page 51.
http://therps.net/User%20Manual/Hawk...s%20Manual.pdf
If the battery wasn't filled and charged correctly initially, it will probably always be flaky. On the other hand, hook it up to a $5 trickle charger from Harbor Freight, and see what it does - it might have just had a low charge the day you tested it from sitting around. In any case, just add the cost of a new battery to the budget, worse case. I wouldn't rank this as a big concern myself.
Hot is pretty relative. The carb may just need to be rejetted if it's running lean. If it's stock, more than likely it's set up lean, and you'll also want to check valve lash. That's kind of expected tuning. You also indicated that the idle is off, so expect some normal carb/tuning work if you do get it, but you'd have to do that anyway with a new bike. It'd be hard to guess if it's running excessively hot without more info, but if the oil is OK and you described the smell as "slightly" not "smoking," I'd guess it's in the normal range from your description.
Still, since the price difference from a brand new one without any unknown history is so small, I'd think I'd get one shipped new just to avoid the extra labor on the electrical issue and spend the time instead on assembly and know that everything is tightened up just right. Whatever direction you choose, good luck. They're a lot of fun.