OK, you're probably going to get a bunch of responses that reduce to "lawyer up." And you could do that. But as a friend of mine (one I consider very wise) once said: "once the lawyers are involved, you already lose, no matter what the outcome."

If you buy that, and if you want this resolved A. as fast as possible, and B. resolved in a way that avoid conflict, drama, and stress, and C. believe that any $$$ is better than no $$$, then consider this option:

Email the client and say "Tell ya what: we both want this resolved as quickly and painlessly as possible. Make it 75% of the invoice amount, and send the check, and we'll call this good." or something roughly to that effect.

Let em dicker a little bit and if you get an amount that you can live with, then take it and let it go.

I wouldn't necessarily resort to threatening to "name and shame" but you can do it a sort of roundabout way if you want, by adding something like "At this 75% number we can both agree to a mutual non-disparagement clause as part of the settlement, so everybody can be confident they will leave this situation with no harm to their reputation, etc." or something to that effect.