How should one deal with money mixed with unlawful (haram) elements due to its deposit in an interest-based (riba-based) bank, given the absence of an Islamic alternative, and knowing that the interest rate varies with the amount, making it difficult to determine the unlawful proportion?

You can refer to the bank to distinguish between principal and interest. If that is not possible, then you must strive to estimate the amount of what is unlawful. Al-Qurtubi said in his exegesis of the verse of riba (usury) (Al-Baqarah: 278) that the way for one who possesses unlawful wealth—if it is from riba—to repent is to return it to its owner. If he cannot find the owner, he should give it as charity on the owner's behalf. If the matter is unclear to him and he does not know how much is unlawful and how much is lawful, he should strive to determine the amount in his possession that must be returned, and give it as charity on behalf of its owner if he despairs of finding him. Ibn Muflih said: "Whenever the amount of what is unlawful is unknown, one should give as charity what he believes to be unlawful." Based on this, you are obliged to strive to determine what you predominantly believe to be unlawful usurious interest and pay it to the poor and needy. If you fear the loss of money, you are excused for depositing it in a usurious bank out of necessity, while limiting yourself to the extent of the necessity, such as opening a current account instead of a savings account.