I doubt that the melted plastic or problem at the mains is connected to or a result of anything going on at the three-way switch. Apparently there is additional work being done at the same time.
There is no doubt that the integrity of the panel has been compromised,, the question is how much and is the HO or person doing the work capable of changing out a panel or willing to make the financial investment?
If he can't get the three way working, maybe he should quit while he is ahead
(no disrespect intended, we were all new guys at one time).
My guess (and it is only a guess) is that during working in the panel, the breaker was disturbed and became loose, which caused the over heating. Replacement would depend on the amount of damage the event caused.