I don't think you even need to go that far. Just refute the charges with your credit card. Very high likelihood of a successful refund since they already acknowledged their error in writing.
I would be that would be highly unlikely to succeed. I have tried to dispute charges with my credit card for similar issues, and they always side with the business. I don’t think I they even check.
Can companies decide not to serve you on the basis of a successful lawsuit you had against them?
If not, then it might be better to go the small claims court route.
doing charge back means Anthropic will ban you forever.
There's a fundamental power imbalance: if you do this to any service, they will likely ban your account. So the monetary reward has to be enough to merit moving all your data and workflows off them in advance and never using them again.