Can I heat one room and cool another with an in-wall AC?
I'm not an expert to tell you whether this is actually a good idea, but an AC unit used for heating is called a heat pump. There are two differences between "a heat pump" and "an air conditioner":


Heat pumps contain a reversing valve which reverses the operation so that what would usually be the cold side gets hot and the hot side cools instead.
The thermostat will be suitable for heating (turns on when too cold) as well as cooling (turns on when too warm), and also control the reversing valve to switch modes as needed.


In your case, you only care about the second feature. Thus, buying a heat pump to be installed through your wall is one possible solution which gives you a thermostat that works as desired without needing external control.

However, any prebuilt unit will also be designed to put the compressor noise on the "outside", which would be your bakery and might be less comfortable than making the proofer room noisy.

You could likely get a more pleasant result (noise in the proofer room instead) by getting a system designed and installed for the exact application by a HVAC technician; it would probably be slightly more expensive, but it wouldn't be particularly harder to install and maintain than a more usual cooling installation, I expect.