He didn't have a personality, but I like to think he was shy (like his performer, Steve Whitmire) except for whenever he was playing the trumpet.
Then again, maybe he was really shy, despite playing the trumpet like a pro, and left the group because his trumpet-playing drew too much attention to him.
The writer of Muppet Snow White did mention in a Tough Pigs interview that he chose Lips to play Bashful because of Steve Whitmire's shyness combined with Lips never developing a personality (by the way, did Lips get any dialogue in that comic?).