All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
If the company that you are dealing with does not own the car and will buy it with a usurious loan to rent it out to you, as you stated in the question, then it is impermissible for you to partake in such a transaction because it implies approving of and cooperating in a prohibited transaction. Allaah, The Exalted, says (what means): {And cooperate in righteousness and piety, but do not cooperate in sin and aggression. And fear Allaah; indeed, Allaah is severe in penalty.} [Quran 5:2]
However, if the company will buy the car through a lawful transaction or had already bought it, even if with a usurious loan, then there is no harm in arranging with this company to rent it out to you for a fixed sum of money with a promise to transfer the ownership to you upon paying all the installments. This transaction is known as 'lease ending with ownership'. It is allowed under Islamic law provided that certain conditions are met and without which the transaction is deemed prohibited.
The resolution of the Council of the Islamic Fiqh Academy of the Organization of the Islamic Conference (in its Twelfth Session held in Riyadh, Kingdom of Saudi Arabia) stating the criteria for permitted and prohibited forms of lease ending with ownership, reads:
"Permissibility Criteria:
1. The presence of two contracts that are totally separate and independent with respect to the time of their conclusion and in which the sales contract succeeds the lease contract or the presence of a promise that would enable the lessee to become the owner at the end of the contract period (as an option for him to buy the leased item). In this connection, Option and Promise are equal with regard to their sharee'ah rulings.
2. The lease contract should be genuine and not just a cover up for the sale contract.
3. The leased property should be guaranteed by the owner and not the lessee. In this sense, the owner should bear any damage that is not caused by misuse or negligence of the lessee. The lessee bears no liability if such damage has made the property useless.
4. If the contract includes insurance of the property, then insurance should be made according to the Islamic methods and at the expense of the owner alone.
5. Throughout the lease period, the contract should be subjected to the sharee'ah rulings on ijaarah (lease), whereas the sharee'ah rulings on sale should be observed once ownership of the property is transferred to the lessee.
6. The cost of maintenance, excluding operational expenses, should be borne by the owner, not the lessee, throughout the lease period."
Only if these criteria are met, the contract in reference is deemed lawful.
If the contract is valid and lawful, then there is no harm in paying the rent or part of it in advance regardless of whether there is a condition stipulated regarding this or not because deferring payment to the appointed term is your right, and you have the choice to waive this right.
Allaah knows best.