I sold a converted tenor banjo recently -- tenor necks are about two frets shorter than the 25+" standard length -- and the buyer was quite happy. I didn't do the conversion on that one, but whoever did not only layered a piece of wood onto the side from the 5th-string peg to the shell (the place where 5-string banjos get wider), but topped it with a piece of fingerboard, complete with frets, despite that no one would be fretting the 5th string anyway. It was a neat job, I have to say.
I long ago did such a conversion, but I left the 5th string just 'hanging out' from the peg to the bridge. I didn't like it, so I put extra wood on the side of the neck, but didn't bother with a fingerboard on top (too much work for what was an old low-grade Kay tenor).
I have a banjo on eBay right now, a 1925 Gibson (10.5" shell -- nothing fancy, in other words), but it has the full-scale neck, so unless you want to use a capo all the time, it's not what you want. However, it is really light, with a good action and sweet, sweet tone (probably close to the plunky tone your teacher is after for you). I used it for several years and always loved the tone. It is not a short-scale instrument, but might be what you're looking for, though it's also probably more than you want to spend.
I have an American Princess, as mentioned above, and it's a lovely banjo, but mine, at least, is only successful with nylon strings, and isn't really loud enough for string-band use. It's the very thing for 19th-Century "guitar style" ("classical") banjo, though.
Good luck in your search.