Is it permissible to deal with a bank that claims to be Islamic and offers a deposit account that neither increases nor decreases, and charges annual fees, while offering other accounts that guarantee the principal with a profit upon earning, and claims to have a Sharia supervisory board and to use Tawarruq, knowing that other banks are usurious or their Islamic branches offer a guaranteed interest?

There is no objection to keeping the account in the aforementioned bank as long as the guarantee of the capital is not a condition in the contract, and the bank only engages in low-risk transactions. If the bank has a reliable Sharia (definition) supervisory board, it is permissible to follow its rulings. There is also no objection to the bank charging fees for an account on which it does not provide profits or interest.