Not sure about the moped style but the first machine that comes to mind is the Aprilia RS50 (Restricted). That's kinda way outside the price range though.

Then there's the Yamaha TZR.

Motohispania has the low powered manual machines.

See if you can find anyone that has an old Honda NS.

The last machine I can think of that probably isn't worth it but may guide you in the right direction is this;

I can't think of anything under $300 that would be street legal unless you make it street legal, (adding turn signals, tires and mirrors) like a cheap dirt bike.

This doesn't answer your question but it does give you some ideas. Good luck.