Is it permissible to purchase equipment and supplies in installments if the provider deals with an interest-based bank, and this bank will transfer the installments in coordination with the bank where the allowance is disbursed?

If the equipment is to be purchased from the owner according to Islamic deferred payment regulations, the most important of which are the owner's possession of the equipment and the absence of late payment penalties in the contract, then there is no harm in that. It does not matter that the supplier deals with a usurious bank, even if the bank coordinates the transfer of installments to them, because what matters here is the permissibility of the transaction itself.