Is a father's bequest of a car to his son, in exchange for the son paying a certain amount to the mother, a valid Islamic will, and does it entail sin?

If it is confirmed that the mother is not making any demands and that the sisters have relinquished their right, then the car is yours. The father is not sinful because he did not intend injustice, and the bequest to an heir is valid if all adult heirs permit it, because it is in the nature of a gift.