How can I clean a B+W polarizing filter that has become so stiff that it's basically unusable?

I see three possibilities, not all of which might be DIY
In terms of delicacy vs price, that's right up there in the "I really ought to take it to a pro" stakes.

1. Dirt has got in it.
   Well, basically, it's made of metal & glass, so as long as you don't drown it, a touch of WD40 might ease it. Spray it onto another surface not onto the filter itself & use a cocktail stick to put the tiniest amount into the joint. Any slight leakage to the outside should be cleanable - inside, not so much.
   I'd advise against anything more invasive. It might physically survive a rinse under the hot tap, but you may never get the inside clean & dry ever again.

However, my money would be on…

2. It's no longer flat.
   Test by placing it on a dead flat surface, maybe a mirror, & see if there's any movement/rocking/visible gap. If it's distorted you'd have to weigh professional intervention against how gently you might try to rectify that.

3. It's no longer circular.
   You can try gently squeezing it from the edge, turning slightly each time to see if it frees up with pressure in one direction, but I doubt that's a DIY fix at all. Take it to a professional.