Buttigieg did Biden a pretty good solid by dropping out of the race. He realized that he and Joe were splitting votes, allowing Bernie to show better. Bloomberg should realize pretty soon that he's running the risk of doing the same thing.
Despite that, I don't think Buttigieg makes a good VP choice. He's still green and considering that Biden is already well into the geezer stage of life the VP has to be ready to step into the top spot. In addition, he comes from a state where the Democrats have virtually no chance of winning. VPs have to be chosen with at least an eye towards pulling another state into the win column, or at least shoring up a purplish state.
Klobuchar qualifies on that score (Minnesota barely stayed blue in 2016) and she's a woman. She's youngish, but not too young. Abrams is another that might make sense; Georgia went for Trump, but only by 5 percentage points. She's black and female, which makes her a two-fer, and which might help boost African-American turnout which the Democrats absolutely need.
Bernie has already said that he will only take a VP who supports his Medicare for All plan, which obviously lets out Buttigieg. The key here is that if Bernie gets the nomination, he's going to know that whomever he picks is going to be the standard bearer for the Democrats going forward. So he's never going to pick a centrist (even though that might be what he needs). Warren? Ayanna Pressley? Maybe Sherrod Brown?