I'm not sure if any foul has been committed as of yet.
The OP said they were producing a sign, but were not installing it. That to me sounds like they are indeed building a cabinet and not installing it. However, maybe they are subbing it out and are going to decorate only the face and the wholesaler is telling them they can get away with a 7" deep can and they don't believe the wholesaler.
I see nothing wrong with a 7" single sided sign.
The part that gets me, though....... how would someone who knows nothing about building these things or installing them land a job like this by bidding it out and not knowing the basics of building one, let alone the costs ??
This kinda chit really gets in my craw and this happens so often around here... and to people who never know what they are doing. How on earth are people landing jobs they know absolutely nothing about ??
It's got to be about the bottom line.
Either someone is trying to sell them a 7" cabinet and they want deeper.... or they want to see if they can get away with a 7" cabinet since they quoted it too low and the wholesaler is trying to talk them out of it.
I say go for the 7" and if you or the your customer don't like it, replace it with a pan face.