Do I need permission to use a food product brand in my finished (different) product?

Not if you are buying that cookie from an authorized dealer of that cookie.

Short answer is: yes, if you plan to identify the branded product as a part of your product. However, if you plan to use the branded product as a base for your product and the customer has to buy the branded product separately, you can refer to it generically. For example, if your product is a decorative desert, you can instruct the customer to purchase "prepackaged cookie dough" and then add your product.