Must the claimant pay the non-refundable insurance excess to the car rental company after an accident caused by another party, knowing that he does not possess sufficient funds for it?

If the error was from the car driver and not from you, then there is no guarantee on you. Participation in commercial insurance is fundamentally impermissible, but it is permissible in cases of necessity, such as situations of legal compulsion or an urgent need to avert extreme hardship and difficulty, especially for Muslims in the West where no Islamic alternative exists. If the insurance is compulsory, you are excused for subscribing to it, and as long as the insurance companies have undertaken to rectify the error, there is nothing upon your companion.