The coding would be extremely simple, and would remain the same for each label, changing only the wording.
So, LCD for ~$6, cheap Arduino nano clones from China (don't knock 'em, they work fine) ~$4, power supply, which could handle several of these units, a few bucks. Some wire, and a container for the parts, ?$.
As cool as this might seem, it's going to cost maybe $12 per unit. If I were going to spend that, I would try to add some kind of useful feature(s). I know! monitor and display the temp of those delicious BBQ beans! Use the second line of the display to thank the cook. Or dispense gastrointestinal advice, as required.