Is it permissible to take a loan from the bank to complete the payment for a sister's house, knowing that the available amount does not equal one-third of the price and that real estate prices are increasing?

It is not permissible to borrow from an interest-based bank, because Allah has cursed the one who consumes usury, its giver, its scribe, and its two witnesses. Allah does not bless anything that involves usury. And whoever gives up something for the sake of Allah, Allah will compensate him with something better. If borrowing with interest is not permissible for your sister and her husband, then it is even more so impermissible for you to do so to help them, especially since borrowing with interest to buy a home is not permissible except in cases of necessity.