Is this an insurance bad faith claim?

The insurance didn't fix the place for you. They fixed it for the landlord who has the policy with them. Now they are attempting to recoup the cost and have every right to do so IF they can. No doubt they are viewing the tossed cigarette butt as an act of negligence. As the tenant, they see you as responsible. I think you should at least speak to an attorney. It may also be necessary to you to sue the careless relative, if you're willing to do so.