Am I obliged to pay my wife's brother the price of his car after it was damaged, knowing that he refuses to take the money?

If your brother-in-law willingly relinquishes his right, he is entitled to do so, because it is a right specific to him, and he has waived it. There is no obligation upon you. If you wish to reward him, that is permissible, for the Prophet, peace and blessings be upon him, said: "Whoever does you a favor, recompense him. If you do not find anything to recompense him with, then supplicate for him until you think that you have recompensed him."