A few possible options:
Is there ANY way to locate the fridge outside the studio rooms... like in an outer hallway, waiting room, garage, basement, etc? I'm sure clients won't mind having to take a short walk to get a cold drink. Many studios I've been in have had fridges located in rooms quite a ways from the audio areas or even in adjacent buildings, etc... not a problem.
Or just keep a heavily insulated cooler in the studio room and when clients come by, fill the cooler with cold drinks right before they arrive... a good cooler will keep drinks cold for a pretty long time... throw in some ice if you really want good performance.
Or... you COULD potentially build some type of sound-cabinet thing around the fridge to help reduce compressor noise, but you WOULD indeed have to vent the fridge... you could build something like those computer isolation cabinet things... basically a big box of fairly heavy mass (a few layers of plywood or whatever), then add some acoustic foam inside, have the whole box "floating" on some foam or pucks whatever, and on the rear panel that faces the wall, have an air inlet port and air outlet port with a small fan or two to make sure you have constant positive airflow to the rear of the fridge. Start with one small fan maybe and monitor temperature inside the iso box... as long as things don't get too hot in there, it should be fine... if heat seems to be an issue, make the air ports larger and add an extra fan or two etc until temps seem ok. If the fan(s) are small and of the quiet variety, plus if they are located BEHIND the iso box and there is some acoustic foam on the back wall across from the fans to eat up any refections off the back wall, the fan noise should be barely audible and way less disturbing than the fridge compressor.
Or, as suggested earlier in this thread, you could simply turn OFF the fridge while doing critical audio work. I personally would not put it on a fancy switched circuit. As another poster said, you cannot switch compressors on and off too rapidly... you're better off just unplugging the unit while doing critical audio work, then just plug it back in later once the audio work is done. A decent fridge will stay ice cold for a long time once shut off... to enhance performance, keep a lot of "cold packs" or ice trays etc in the fridge so that when you shut it off, you have some extra cooling momentum. Only problem with this idea is that when you shut off a fridge, you could possibly encounter excessive "defrosting", any residual icing in the freezer section etc may start to melt and leak... it all depends on the condition of the unit, etc..... you could always try it and see how it goes. If you do unplug the fridge, just don't plug it back in for at least 5 minutes or the compressor will be unhappy.
But the easiest solution is to just get the fridge OUT of the audio rooms. In the place I'm currently working on, there will be NO fridges or coffee machines etc in ANY audio rooms. For me, expensive audio gear and food / drinks do not mix. When it's snack time, the client can run out to 7-11... or there's a nearby kitchen that we can visit during a break to get hot or cold drinks. For a busy session, I might just bring a cooler with cold bottled waters so everyone has very easy access to cold waters. Avoid the fridge if you can... remember, any fridge pulls a lot of juice too... who pays your electric bill?