Is it permissible to initiate a contract with an interest-based bank and then transfer it to an Islamic bank due to the unavailability of a 20% down payment for purchasing a home?

What the questioner described as "starting a contract with a usurious bank" means borrowing from the bank with interest. This is forbidden usury (riba) and is not permissible except in cases of dire necessity. The purchase of a house does not constitute a necessity that permits usury, as long as a person can rent a house without unbearable hardship, which is usually the case.