How about a P/J, or a G&L. The L2000 is definitely worth a look and a play. A Thunderbird is definitely a good one to try.
I would highly recommend a Ric 4003, though, I'll caveat that with "try before you buy". Some people "get" them, most don't. If you're playing a P bass anyway, the neck won't feel too weird (if you played a J bass, it will feel very weird!), though you might need to stretch your budget.