Well in that case. Puch maxi, Honda PA50II(hobbit), Vespa bravo or grande, or any Italian bike with a minerelli v1 engine would suit you well. Simple, pretty reliable, easy to work on.
Hell, you should be able to find a nice one for a little over half your budget. Remember alotta manufacturers just made engines. So you can get a v1, or a Sachs on a general, a Puch e50 on a jcpenny. It's really the engine that defines the bike, whereas the frame and Vin could be from any number of manufacturers.
Make a post in buy/sell forum looking for a sorted unmolested bike. Or one with a few tasteful mods. I wouldn't rule out a bike with an expansion chamber exhaust for example. Just don't buy a modded out moped for your first bike.