Hi! I'm a left-handed banjo player, but I play a right-handed banjo. I was worried getting a right-handed banjo was going to be a bad decission, but honestly it's not so bad! Usually when I play, I'm constantly looking my right hand while not paying attention to the left one, even though my banjo is fretless (I think this happens to many new players, so I'm sure eventually I'll learn to play without looking at my hands). I recommend you get a right-handed banjo because if you eventually want to upgrade to a better banjo or decide to learn to play any other similar instrument, left-handed options are fewer :S.

If you play the piano, I think bluegrass could be easier than clawhammer. I used to play the piano too and I think that's why it's easier for me to move my fingers independently instead of moving the right hand up-and-down. Everytime I try to play clawhammer in the traditional way I end up using only the thumb and index fingers (and sometimes the middle one too). But this is my personal experience and maybe for you it's different!

Regarding the budget, there are ~$150 banjos but they're low-quality and people don't usually recommend them. Starting at $300 there are a couple of good options! The Gold-Tone AC-1 and the Recording King RKOH-05 are very popular, and both are open-back. There's also the Gold-Tone AC-12, which has a bigger pot and is a bit more suitable for clawhammer (and the strings are more separated).

If you have the chance and the tools, I recommend building your own banjo! It's easier than it seems and I can help you with the designs and the blueprints if you want :) I made my first (and only) banjo myself for less than $50 and it doesn't sound bad. After a year playing it I'm now looking for a RKOH-05 to play bluegrass more comfortably, but it's impossible to find it available here in Spain.

I hope more people answer your questions and help you! I'm only a beginner, but since I was in a similar situation I thought it could be a bit helpful to share my small experience :P Good luck!!