shop refrigerator operation

Just about any mech/ electric stat that resembles that one should work. You have the basics correct. The stat controls evap temp. The bleed air door changes the amount of cooling to the refer. Make sure the fan works and is not dragging, and make sure the defrost stat is not sticking if it is still a mech/ electric clock motor type. The only other important component, generally is a defrost termination stat, which stops defrost when the evap warms up. So if it's sticky you can have problems. Make sure the evap drain is not blocked

Probably the biggest other problem is that the aluminum evaps can develop slow leaks almost impossible to detect. These leaks of course, slowly reduce the refrigeration capacity and efficiency. If you are sure this is the problem, and if it is slow enough that "let's say" you get a year out of it, you can rustle up some refrigerant and either weld or clamp on a saddle valve.

Charging these is a VERY iffy deal. Used to use a charging cylinder, but you must take into account the refrigerant in your hoses. They can be very finnicky.

If this is kind of a junker, and beer only, you might consider just really opening up the damper so it can really move some air from freezer to fridge. It might even in some cases be helpful to add a second or install a larger fan

Having worked on my share of commercial / restaurant equipment, it is a crying shame the poor crap us homeowners get