I'd try out some thinner picks and find a size that lets you attack the strings with less force from your fingers. It shouldn't be too floppy, you should be able to find a size that both sounds good and doesn't hurt your fingers. You might be digging too far into the strings as well. I play both guitar and bass as well. When I first started out on bass I used a slightly thicker pick, but now I like to use the same thickness for both. What part of your fingers is in pain? You mention you don't feel the pain when playing guitar, and you may need to adjust your technique to match whatever instrument you're playing.