This is a tough one.

I am a pure mathematician. Early in my career I was in a situation akin to the one of Postdoc B. (Except I was still a master student and the person in the role the postdoc A already had a permanent position. He worked on a problem for a year but had only an incorrect proof. One more thing to add here: I have learned a lot of research-level math from A as an undergraduate student.) Here is how my advisor (who was also a former advisor of A) handled the situation:

He first talked to me and said something along the following lines:

"Moishe, you are brilliant. We all know this. In your future math life you will prove many remarkable theorems. [At the time I felt embarrassed by all this undeserved flattery. In retrospect, my advisor used the right approach here by “smoothing my feathers” first.] You solved easily a problem that A was working for a year. What do you think of publishing the result as a joint paper (with A)? To you it probably will not mean much, but to A it will mean a lot."

He was right, I said yes and published the result together with A. I did not resent my advisor's advice; I also do not regret following it.

Will this strategy work in your situation? I do not know, but, I think, it is worth trying.

I wish all the world were as generous as the situation and solution described in the answer by Moishe Kohan.

However, before you do anything, you need to verify the solution of B to be sure that it holds up. But, assuming it does, there is no real solution but to "bite the bullet" and deal with A. It should be your task, not B's to break the news to A. You will probably also have some responsibility to A to get them to a success point.

But talk to B first to explore their position.

I don't think that A has an automatic authorship position here, given your description, but I also don't think it would be exactly a "gift" to include them.

But, if B insists on sole authorship, as is probably valid, then you need to sit down with A to break the news. You can then explore how much is salvageable from the work already done by A.

And, don't try to pressure B on a joint publication. (Caveat, a somewhat different scenario might call for a very different solution.)

One option in a joint paper is to describe both approaches and why the one is superior. That would make joint authorship valid by anyone's (I hope) definition.

Dilemma appreciated. I'm a sw developer (40 years) who has worked with many scientists & done a lot of collaborative design in different fields, with oft-touchy people.

Further to the excellent answer by Moishe Kohan is the question of framing - would B have thought about A's problem in the same manner if they didn't know:

1. A had explored a lot of approaches
2. a nearly-acceptable answer had been found but accuracy of their solution is not quite where we want it?

That context of A's work as a preface to B's insight is further reason for a joint paper.