I think I would be able to stick a multi inside and control the temp fairly accurately with the GE Outdoor plug. My only concern the Multi can’t survive in a <40° and kind of humid environment.
I’d be surprised if it would. He-he 
I’d recommend a type K thermocouple for this application.
I’m not sure if it’s recommended, but I’ve had a multi in my freezer for the last week or so. I’m using it to keep an eye on the kids leaving the freezer door open like they normally do once a month or so. It’s still working for now at least. 
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