You can only deduct the mileage allowance for a car you own, not one you rent or lease. If you rent a car for mystery shopping, you can deduct the rental expenses and the gas you actually pumped into the car. Mileage doesn't matter.
When I travel on business and fly into a town and rent a car, I deduct my airplane ticket, the rental car cost, and the gas. Whatever I paid to get where I'm going is deductible.
It's important to know whether a particular gas charge on your credit card statement went into the car you own or the one you rented because they need to be accounted for separately if you are calculating actual cost on your personal vehicle.
(BTW, if you use more than one personal vehicle for work, you need to track your miles for each vehicle separately, and if using actual expenses, track those by vehicle too. You may find one car should use actual expenses but the other might justify taking the standard mileage deduction. And yes, it is possible to change methods under certain circumstances. You are not necessarily "stuck" with the method you used first.)
Time to build a bigger bridge.