Is it permissible for a second bank to pay off the financing on my behalf, and is it permissible to transfer the Istisna' contract or convert it to any Shariah-compliant formula in a non-Islamic bank?

There is no objection to building houses through the Istisna' (manufacturing contract) if the description is clear, eliminating ambiguity and preventing disputes. However, if the Istisna' contract has been concluded, it is not permissible to sell the remaining debt owed by the client to a third party, whether it be a bank or otherwise. Selling a debt to someone other than the debtor for a deferred price, whether of the same kind or a different kind, involves usury (Riba). This is known as selling debt for debt (bay' al-kali' bi al-kali'), which is forbidden by Islamic law. An exception to this is selling the debt to the debtor himself for an immediate price.