Is it permissible to perform Hajj with a bank loan, knowing that all financial transactions are conducted with this bank, and if not, is it permissible for my wife who has no monthly income to do so?

A poor Muslim or one with an income is not obliged to borrow for Hajj, because he is not able. Borrowing from a bank is permissible with legitimate Islamic controls, the most important of which is that it should not involve interest. Usurious borrowing is forbidden. If you deal with a usurious bank, you must search for a bank that does not deal with usury. If you have already borrowed with usury, then Hajj is permissible with the borrowed money, and you are not obliged to pay the interest; rather, you are only required to return the principal amount if you are able.