Is it permissible to appoint a non-Muslim lawyer to obtain compensation from the insurance company of the party responsible for an accident, knowing that the lawyer may obtain a large financial compensation that exceeds the value of the actual damages?

It is permissible to appoint a lawyer at the company's expense if it prevaricates and fails to fulfill its obligations, provided no injustice is done to it. As for the costs of the accident and treatment, it is not permissible to take more than what was paid to the company throughout the insurance period, because the excess is usury (riba). And insurance here is compulsory; were it optional, it would not be permissible.