First, I suggest if this cannot be resolved relatively quickly, you take steps to convert the joint tenancy to a tenants in common style holding.
You also need to understand that as long as there is a cotenancy, each of you have an equal right of use of the property. You cannot exclude him without incurring a liability for half the rental value of the home owed to him.
If the payment of $10k to him would result in him relinquishing his ownership interest, I would surely start crunching the numbers. If this ended up in court (a partition action), it’s likely he would recover his down payment money given it is sourced from leveraging his car.