Is it permissible to use a grant equivalent to the value of interest to pay off an usurious bank loan taken out of necessity for housing, and does that make the loan permissible, knowing that the lenders are public entities?

Usury is one of the gravest sins, and it is forbidden to deal with it with any party. Its consequences are dire in this world and the hereafter for those who do not repent from it. It is not permissible to take interest-bearing loans, for your need for housing is not considered a necessity that permits you to borrow with usury, as long as you can find other means of housing, such as renting. As for what the questioner mentioned about his neighbor, if he means his trickery to reclaim taxes by increasing the price of goods for the public institutions he deals with, there is no harm in that.