Aristocrat Cabinet Humidor, will be going....

[shilala]

There it is.
You're only providing cooling for the peltier units (condenser side).
It'd be far easier to use a standard refrigeration unit with the condensing section outside.
Mind ya, this is my opinion, building peltier units into the wall is a very bad idea.
You'll need to allow access.
Then you'll need to duct cooling air across them.
I just wouldn't do it, but if I did, I'd simply box the peltiers between studs, allow access panels on the outside wall, and power a low voltage fan to pull cold air from the floor and blow it out the top.

If the unit can be easily rolled out for service, all that much easier. It's simple at that point. And then I'd almost be okay with the peltiers.
But not for me.
That's waffly, I know.
I'd just use proper refrigeration. The price vs. trouble is inconsequential, but I can do it myself, so there's that.