Is it permissible to give a gift to a person who bought a house with a usurious loan and to accept his invitation to a party celebrating this occasion after advising him and his refusal to back down?

If your relative insists on the sin of usury (riba), you should neither accept his invitation to a house-warming party nor give him a gift on this occasion. Rather, he needs someone to console him, not congratulate him. Usury is one of the gravest major sins. Allah has threatened those who consume it with destruction and war. The Prophet (peace be upon him) cursed the one who consumes usury, the one who feeds it, its scribe, and its two witnesses. He also stated that one dirham of usury is worse than thirty-six acts of fornication. You must continue to advise him to repent to Allah.