I think B only want to see the letter in writing so that they can go slightly above what A has offered.It makes no sense to go largely over. Why not ask B to m

I think B only want to see the letter in writing so that they can go slightly above what A has offered.It makes no sense to go largely over. Why not ask B to make a blind offer based on the value you can bring and what you're worth, tell them you're not interested in them upping A's offer, just formulating their own based on value not competition. You want to hear what they would have offered without company A in the picture. Not only do you come across less money-motivated but I think you're more likely to get a higher offer from B this way. Plus if you do get company B's offer in writing - maybe you can take that back to A. After all if you prefer company A, you should be going with them regardless.