What should one do who works in an interest-based institution, is displeased with his work, and cannot leave it except after finding another job due to his family responsibilities?

The basic principle is that it is forbidden to work in usurious institutions, because usury is one of the gravest sins.

It has been authentically reported from the Prophet (peace be upon him) that: "He cursed the devourer of usury, its payer, its scribe, and its two witnesses," and he said, "They are all alike."

It is not permissible to work in them except out of necessity, provided that this permissibility ceases when the necessity ceases. Allah Almighty says: "And whoever fears Allah - He will make for him a way out and will provide for him from where he does not expect. And whoever relies upon Allah - then He is sufficient for him."