Outside menus are typically now flatbed print with thin plastic inserts I want to say it's really thin lexan, but I'm up really late. They are cheap and easy to change out the cost for an outside digital display is very high for a menu. Other option is you in case a TV but it doesn't have the NIT value that you need. I am interested in what you find because I do have a vendor that sells 6 mm LED signs but they're god awful expensive.
We typically flatbed print onto white acrylic. My concern with changeable inserts is dirt and water will get in there. I tell customers they need to try to limit their updates to once a year or so to get it done at 1 time. Any kind of LED digital type sign is going to be way expensive.