You can enter a billable expense for your mileage since your customer pays you for it. As for the transaction itself, you can use the Category Details (an expense account) directly, instead of a product/service. Don't forget to tick the Billable box, as well as enter the specific customer you want to bill for the expense under the Customer/Project column. As for which class to use to track these billable expenses, you'll be the one to decide. You can either create a new one for this specific purpose or use an already existing class.
See the sample screenshot below for reference:
Open this article to learn more about this process: Enter billable expenses. It includes a short video on how to create a billable expense within QuickBooks Online (QBO).