No way I would pay. Sets a bad precedent for any future litigation from them or others. The whole thing revolves around causation. That panel functioned for nearly a year, passed inspection and only failed when a different electrician was working on it energized. (The duct tape strikes me that he was up to something squirley and trying to protect that edge for conductivity). All this introduces reasonable doubt (about causation), which is fatal to a claim like this.
Especially when you were not notified nor given a chance to inspect or repair. The panel being replaced immediately, any destruction of evidence, makes it impossible to prove whether this was an original installation issue ‘or’ a manufacturing defect. Again, from a liability standpoint, that's a very weak claim against you.
And paying to make it go away when no fault has been established yet seems at the very least premature. I definitely would involve the manufacturer as others have said here. Seeing hundreds of these photos weekly they most likely will have insight beyond yours. And as far as I'm concerned that email is evidence against the last electrician.
*edit: - besides they're just fishing, otherwise you would've gotten a letter from their attorney.