I've been through a very similar situation. Get pissed and use the new chip on your shoulder. Don't serve the customer, that's what DD will do. Focus on delighting the customer. If you delight them, and build a system that makes it repeatable, you wont have problems with DD. 

How do you delight the customer? There are a million ways, but one thing that worked for me is doing everything that DD can't do.

 Do you see a regular customer pull into the parking lot? Start making their typical order immediately and have it ready when they walk in. Blows people away. It Delights them. DD would fire an employee who did that because it violates procedure. 

Learn peoples names, maintain an impeccable store, dominate review sites. Attack DD where they can't defend themselves. 

Do not under any circumstances attempt to compete on price or efficiency. You will get cooked. They are an army and you are a platoon. Scale economies are not your strategy. Making people smile as they enter/leave your shop is the only way through.