Is it permissible for an employee to take money that his manager promised him from the company without its knowledge, given that the company is well-off and he is able to do so?

If you were not negligent or excessive in driving the company car, then there is no guarantee upon you, because it is a borrowed item (ariya) in your possession, and a borrowed item is not guaranteed if it perishes for the purpose it was borrowed for without excess or negligence. Specialists should determine whether you were negligent or not, and if it is proven that you were not negligent, then you have the right to demand the paid amount from the company. However, if you are liable and the company promised you a reduction of half the amount, then if they fulfill it, it is good; otherwise, it is not permissible for you to encroach upon the company's property.