All perfect praise be to Allah The Lord of the Worlds and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
According to Islamic Jurisprudence, it is permissible to receive a compensation against harm. Jurists have clarified the reasons that obligate receiving a financial compensation, but there is no room, here, for further elaboration. In fact, there are many an evidence in this regard. For example, Allah, The Almighty, said, "and pay compensation to the deceased’s family..."{An-Nisa`, 92}. In this verse, Allah, The Exalted, has ordained paying a compensation in case of accidental killing. In principle, compensation for harm is the liability of the doer and could be that of the person causing the action, because what obligates paying the compensation is the action itself. Al-Kasani (May Allah have mercy on him) said, "Undoubtedly, destruction is what obligates paying a compensation, because when something is destroyed it can`t give the benefit it used to, and this is transgression and harm.
Allah, The Almighty, said, "If then any one transgresses the prohibition against you, Transgress ye likewise against him.'(Al-Baqarah, 194)" {Badai` Al-Sanaei`, 7/164}.
If the case is exactly as explained in the above question, the compensation for harm is the liability of the driver represented by the insurance company. Also, it is the right of the injured person to claim the amount, which isn`t covered by the insurance company, from the driver alone, because the car`s owner wasn`t the doer nor the agent causing the accident. Therefore, it is permissible that the lawyer continues the court procedures and whatever amounts he manages to collect are to be given back to the owner who expressed his willingness to pay the whole compensation amount in advance to avoid any legal accountability. And Allah knows best.