Is it permissible to take a sum for car repairs from the other party if the percentage of fault has not been officially determined, knowing that the other party insisted on paying the amount in cash and not waiting for traffic procedures to avoid the damage that would befall him for his violation?

It is permissible for you to take the amount and benefit from it, as long as you are not certain that you are at fault, and your companion admitted to being at fault himself to avoid the traffic investigation, and you both agreed on the amount. This is similar to the permissibility of reconciliation and absolution from a lawsuit in order to avoid an oath and put an end to a dispute. However, if one of the parties to a reconciliation based on denial lied, then the reconciliation is inwardly invalid for him, and what he took is unlawful for him, according to the Almighty’s saying: "And do not consume your properties among yourselves through falsehood." Since you do not know that you are at fault, and the man does not want an investigation and is content to bear the fault, there is no harm upon you in accepting the reconciliation and benefiting from what the other party offered.