Christine,
I am primarily a bari player and I have a couple recommendations that work for me. I am a big fan of the vintage style metal berg larsens. When I was in high school about 10 years ago my jazz band director lent me his personal berg 95/1 metal piece which was a good mouthpiece for contemporary big band, R&B, and funk. I now play my own 120/1 berg which I wouldn't trade for anything and play on all styles. If you're looking for a more traditional big band type sound, I play a Jody Jazz HR* on both tenor and soprano ~$120 and love the rich, classic sound they produce although I've never tried one on bari. They play similar to a hard rubber Otto Link but much, much better and a lot more consistent. I don't think Jody makes any bad products, it seems to me that all of their stuff is really good, although some of their products like the DV series are quite pricey, but worth it if you can afford it.
Another one to try might be the Brilhart Levelaire mouthpiece. It is a metal piece with a low baffle and open chamber for a nice big sound, but it comes with a slip in spoiler that accentuates the high overtones and gives it a brighter pop/funk sound. I play one with the spoiler on alto and like it a lot for pop music. They are also very reasonably priced.
I like to order from wwbw.com (the woodwind and brasswind) they will let your order and try three mouthpieces at a time so you can test and compare and then return the ones you don't like. I find it a great way to shop if you're not near a music store with a large inventory. It's allowed me to really pick mouthpieces that give me the sound I'm looking for and fit my playing style. Good luck to you.
