Is it permissible to open two accounts in an interest-based bank, one for daily transactions without interest, and the other for savings with interest, if the bank stipulates that as a condition to protect the primary account, especially since dealing with interest-based banks is considered a necessity in non-Muslim countries?

It is not permissible to deposit money in interest-based (riba-based) banks except out of necessity and in the absence of Islamic banks. In such cases, it should be limited to a current account without interest, as a way of choosing the lesser of two evils. Opening a current account in an interest-based bank is forbidden because the bank utilizes the money in its illicit activities. However, if a person fears for their money, then depositing it becomes permissible.

If keeping money in a single account poses a risk to it, then there is no objection to opening another account—without interest—after verifying the bank's claim. And if the risk can only be removed by opening a savings account, then that is permissible, provided that the interest is not utilized but rather disbursed for the benefit of Muslims. One should verify the existence of the alleged risk.