I think with a hard $200 limit you're not going to find anything new, at a bike shop or online.

Honestly, I'd look for an old rigid mountain bike on craigslist. Look for something listed as around 15 inches and made by Trek, Giant or Specialized. If you find one of those and it is in decent shape (i.e. all the moving parts move smoothly and nothing is rusty) you should be OK. You should be able to find something like that for around $100, though you might need to drive to a larger city to get it.