Is it permissible to sell an apartment that was purchased through a usurious bank with the aim of buying another apartment through an Islamic bank, especially given the family difficulties and the children's desire to live in an upscale neighborhood?

Your engaging in a usurious loan is forbidden, because the necessity that permits a usurious loan is when the individual reaches a point where, if they do not consume what is forbidden, they will perish or be close to perishing. Buying a house and owning luxuries is not a true necessity. As long as you can find housing, even if through rent, that is not considered a compelling necessity for usury. You must repent to Allah Almighty for what you have fallen into and rid yourself of the usurious loan as much as possible, without being obligated to get rid of the purchased apartment. It is permissible to deal with an Islamic bank if it adheres to the Shariah guidelines in its transactions.