We just did a couple signs exactly like this. We printed on the back in reverse, then applied frosted vinyl to it.
They dont want customers to pick at it? Front piece will be glass reverse printed, or applied vinyl if you'd rather. Cut a second piece of thin plexi, or whatever you want. thats about a quarter, or half inch shorter all around (Same as what it looks like in your picture), frost it.. and attach it to the back of the glass with the standoffs. Then the customer can't pick at it.. and it'll give it a slight shadowing look.
We did one for a hospital just like that. Reverse cut vinyl / reverse print clear... Middle piece was frosted vinyl on thin plexi, Then back was the same thing. Looked great, and nothing could be picked at.Making sure everything was perfectly aligned on both sides so there was no shadow was a bit of a pain... But since yours is just printed on one side, that wont be a problem
If they don't like this idea, which solves all their problems... Tell them you can't help them. It's not worth it for such a small sign, and their expectations are too high.