Is working at an accounting firm whose tasks include paying clients' credit card bills before the due date to avoid late fees a mix of what is permissible and what is forbidden? And is it permissible to continue in it until finding another job if it is dubious or forbidden?

The work you mentioned at the company, where the client tasks you with paying his monthly bills, and his use of the credit card through his bank account before the final payment due date, does not appear to involve any impropriety; as there is no direct involvement in the unlawful, nor is there assistance in it.