Is it permissible for me to request a car rental allowance from my employer, knowing that the nature of my work does not require the use of my private car?

If you honestly describe the nature of your work without deception, and the responsible management agrees to grant you the allowances and bonuses, then there is no blame upon you. The crucial point is the administration's acceptance and that the request does not involve deception or trickery. The Prophet, peace and blessings be upon him, said: "Whoever cheats is not one of us," and "Cunning, deception, and treachery are in the Fire." Allah Almighty has warned against consuming people's wealth unlawfully.