We can discuss the structural issues and the customer's preferences, but from a business perspective, there is a "I don't want to be sued out of my socks" perspective. I'd explain that your design (2 doors) is structurally sound. One door without some sort of mid-shelf (shelf material?) support presents safety issues that can only be mitigated with safety glass, extra hinges, and a signed waiver of responsibility. There will be extra cost involved. If your customer is good with that, stop thinking about what they will pay, give them the whole picture and true costs. Also explain that from your builder's experience perspective requires you to have them sign a waiver and disclaimer of any future issues with that cabinet. Also tell them they likely can find someone to make as they wish who will not be concerned with the safety issues, but the recourse could have a price to them in injury and/or destruction of treasured items that likely will require court action to try make things right.
Look at your motto. If in doubt (and you certainly are here), build it stout.
They will respect you in the morning.