Is it permissible to engage in a borrowing transaction from an investment company to pay the price of cars being purchased, in exchange for paying a fixed amount to the investment company as a transaction fee, regardless of the car's price?

Borrowing with a condition of increase is forbidden; it is usury (riba) whose prohibition is a matter of consensus, no matter how small the increase.

Every loan that stipulates an increase is forbidden, and every loan that brings a benefit (to the lender) is usury.

If the company lends the price of a car, on the condition that it recoups its price with an increase of $200, then this is forbidden usury.

The increase is not considered a loan processing fee unless the processing fee is the actual cost of processing the loan; anything beyond that is usury.