Is it permissible to finance a project through a bank with a Murabahah contract, given that the bank deals with Riba (usury) in addition to Islamic transactions?

There is no objection to financing the project through Murabaha, provided that it adheres to the Shariah conditions, which are: the bank truly owns and takes possession of the commodity before selling it to you, the contract is free of a penalty clause for late payment, and any previous contractual link between you and the original seller is nullified. It is permissible to conduct this Murabaha with a usurious bank, because dealing with a usurer through buying, selling, and other inherently permissible transactions is permissible.