Offer to keep the menus up to date for them yourself, for a price. Now you know how much they’ll be willing to pay for a solution, and you’re in a nice position to develop your product from (that will replace you).
This approach is also forcing you to interact with potential customers. The value to you of those relationships is both quantifiable in terms of revenue but also unquantifiable in terms of learnings. Do as much as you can to learn about your customers and your product before spending anything on building it. This is the definition of MVP.