You should be able to get a good open back banjo for your budget or less. Think about going second hand. Unlike guitars, most open back banjos are a bolted together collection of bits, easily repaired, adjusted and upgraded.
Unlike a guitar you don't need the "perfect" setup for a banjo to be playable. And I really wouldn't worry about "tone". All the open back banjo players I know (myself included) stuff rags between the neck stick and the skin to get the tone they want. Whether a $5000 banjo sounds "better" than a $500 banjo is a very mute point!!!
All the recordings in my signature were done on a 1910 banjo, that's probably worth $500. And I gigged recently with one from 1895 worth about $150 max and a Gold Tone one worth about $400 second hand (in the photo).
With a bit of tinkering you can get great timbre and playability from just about anything!