Is it permissible to purchase a house through the British Al Rayan Bank, which describes itself as Islamic, knowing that it determines the rent based on the percentage set by the Bank of England?

If the bank engages in a diminishing partnership with the customer, or in an Ijarah Muntahia Bittamleek (lease ending in ownership) contract, and the bank determines the rental value based on what was mentioned in the question, this does not affect the permissibility of the transaction. The important aspects are the clarity of the rent, the customer's consent to it, and the compliance of the transaction's terms and conditions with Sharia principles.