Is the action I took—accepting that the driver of the car who caused the accident was another person who had insurance, and changing the car's position before the traffic police arrived, with the aim of helping the at-fault party who did not have insurance avoid imprisonment—religiously permissible? And what should I do if it is not, knowing that my insurance company has demanded the cost of repairs from the other person's insurance company, and I feel guilty for committing what might be deception or bearing false witness?

Commercial insurance is prohibited. If one is compelled to engage in it, they are excused to the extent that an compelled person is excused, and the sin is on the one who compelled them. If one cannot avoid it, they may insure for the minimum legally acceptable amount. It is not permissible to exceed this to comprehensive insurance unless no other option is allowed. The sin of bearing false witness is very grave. If what you did was motivated by an attempt to relieve a Muslim from distress, and you did not foresee the outcome, then we hope that no sin will befall you. As for compensation, if it is possible to avoid it in a way that makes the entity that was supposed to pay responsible for the payment, then that is what must be done. But if it is not possible except by revealing the truth, and that entails harm for you, then it is obligatory for you to bear the cost yourselves, as this is choosing the lesser of two harms.