Is it permissible for a partnership to be formed where capital is provided by one party, effort is divided between the two partners, and profit is split into three shares: one-third for me, one-third for my partner, and one-third for the car used in the trade? If not, what is the correct way to handle such a transaction?

If what is meant by a third of the profit for the car is that you both agreed that your partner gets one-third of the profit, and you get two-thirds of the profit (one-third for your contribution of money and effort, and one-third for using your car in the business), then this transaction is permissible according to some scholars. It is mentioned in "Bahr al-Madhhab (definition)" by al-Ruyani that there is a saying permitting this if the owner of the car intends that what is for the car is for himself.