The idea that Biden would or should pick a Republican is just hilarious. Should Trump dump Pence for Doug Jones? Of course not. There are always more votes in the base to protect than there are in the middle up for grabs.
Party-switching in presidential elections is somberly predictable. About 4 or 5% of self-identified Democrats voted for Trump, just as they did for Romney, McCain, etc., and about 4 or 5% of self-identified Republicans voted for Hillary, just as they did for Obama. Any extra beyond that, if Biden is going to steal them, he's going to do it because Trump is Trump, not because X is his running-mate. And the truth is, when the dust settles, it will probably again only be 4 or 5%. Picking a Republican as the Veep won't gain him anything.
But it will lose him plenty. Lots of liberals simply stay home or protest vote if he goes that route. Biden's voters are demanding four attributes for his running-mate: Democrat, woman, black, liberal. He's probably not going to hit all four. There isn't a whole lot of overlap between black and liberal, and that's the main reason Warren is still in this thing, and there's so much consternation over the eventual choice. If Warren were black, she'd already be announced, just as Harris would be if she were a darling of the left.
But he's gotta hit three of them, at least. Whether he goes black or liberal, he'd better pick a female Democrat. No getting away from that.