I know this goes without saying, but now is the time to have a team meeting and stress the importance of product quality and customer service. I would also take the opportunity to produce an in store poster or two (if you don’t already have them) which emphasize your strengths (market differentiation such as quality ingredients, “old world” style, etc.) as well as your connection with the community. This doesn’t have to be sappy but you need to help reinforce the notion that you’re “their local pizza shop”.
Yes, you’ll see a dip in sales. And you’ll probably lose some sales permanently. But your commitment and quality will keep you in the game and ultimately, make you a better pizza maker!
But while you may not be able to put them out of business, your running a strong race and keeping your market share may keep them from being as profitable as they’d like to be.
Don’t get in a price cutting war as you will probably be the loser and your loyal customers will leave you after your quality and service drops.
Good luck!