Under $200, for a good selection, you'll have to go vintage or wait for Flea to come out with their cool new banjo uke, which will list, we're told, for about $220 with a wooden fretboard and under $200 for one with a plastic fretboard - though that doesn't have a resonator.
There is only one banjo uke in production that lists for under $200, that's the Rally, which comes in resonator and open back models. I'm not a fan, but haven't played it enough to have more than a superficial opinion -
http://cgi.ebay.com/Ukulele-banjo-4...989?pt=LH_DefaultDomain_0&hash=item53e5b02595
For both vintage and new, I'd recommend Elderly instruments to you - check out what they currently have:
http://elderly.com/search/elderly?terms=Banjo+(180U+or+180N) Of what's there, I can recommend the La Venezia, which is in your $ range and is a good quality J.R. Stewart instrument. Slight problem is that its an open back, but J.R. Stewarts were high quality inexpensive ukes and the guys there will play them for you over the phone or skype so that you can get a strong sense of condition and sound. I love these guys at Elderly.
I'd also recommend that if you go the eBay route, that you look at Slingerland "May Bell", either Model 20 openback or Model 24, which is a flat backed resonator instrument. Both are in your price range, and they come up more frequently than any other banjo ukes - often under the S.S. Stewart "Collegian" brand for the model 20 and "University" brand for the model 24. They are also sold as "Winner", "Victory" or other brand, and often, they have no label and are misidentified by sellers.
On eBay right now are the following, which fit your requirements:
A William Lange ukulele - not identified as such - which should go for about $200:
http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=250757702469&ssPageName=STRK:MEWAX:IT
Another Lange, which is identified by seller as a Vernon (Lange made these for Vernon, though this one appears not to be one of those as there's no indication of a brand plate). This one looks like a winner and can probably be had for not more than $150, but you want to make sure they can tell you if the neck is straight:
http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=330519089606&ssPageName=STRK:MEWAX:IT
A Slingerland Model 20, which is a GREAT little instrument. This particular one will needs more work than you may want to do, but this is a model that you should perhaps look for as it's probably the best cheap antique banjo uke for playability and cost:
http://cgi.ebay.com/Maybelle-Slinge...980?pt=LH_DefaultDomain_0&hash=item2a0d67447c
A J.R. Stewart "Le Domino", which is a great instrument, worth about $350, but it could always go for less, given the small number of buyers looking for a Le Domino:
http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=320643246636&ssPageName=STRK:MEWAX:IT
Finally, I'm selling the Slingerland Model 24 in this video:
http://www.youtube.com/user/MrJnobianchi?feature=mhum#p/u/0/EUuE8JMwz0I
It's in VG+ condition in terms of the metal and the pot. There's a slight issue in that the heel is warped, but happily, that doesn't effect the fretboard, as you can see. I did a light restoration with a new goatskin head, new Grover tuners and bridge, fully cleaned and polished and the neck re-set. It's got a flat resonator back, a brass tone ring, and it's a great little player that can be played very loud. The only reason I'm selling is because I haven't got the room, so it goes for less than I put in - $170 + shipping.
Let me know if you're interested, and even if not, I'm always happy to look at anything you find, as are all of us banjo-ukers.