What is the ruling on depositing money in a bank that claims to be Islamic, while it is not, especially with the unavailability of a permissible (halal) alternative for preserving money?

Islamic banks vary in their adherence to Sharia controls. It is permissible to deal with those that are compliant, and a transaction is not forbidden based solely on suspicion. If there is a concern for one's money, and there is a need to keep it in the aforementioned bank, then there is no harm in that. Islamic banks, even if they have some violations, are better than riba-based (usurious) banks.