My experience as an installer and service tech for fountain equipment...fortunately has never provided me with this type of situation...
Although It sounds like a fun challenge.
What is the make / model of your fountain unit?
Also, what type of valve....this may be hard to determine....but a good description may work.
The soda fountain design provides the absolute Ideal condition to create good quality soda....The machines like you describe are very efficient at pulling the heat out of the soda water and syrup as it passes through the Ice chilled cold plate. To create good quality soda, the temperature of the product must be below 4o degrees....ideally 38....as it mixes together at the valve and enters your glass....that's with out ice. If it's above...it will be foamy and flat....the warmer it is the worse it gets.....even if you pour it over ice.
I think the keg Idea with the mini fridge may be the lesser of 2 evils. I would plumb the soda out port from the carbonator to a Corny keg inside the fridge....From the corny keg back out to the soda in of the unit... that would give you a 5 gallon reservoir of cold water.....not cold enough by the time it mixes with the warm syrup at the valve because the syrup won't be chilled...but better than nothing...If it's just you....it may be just enough to be work.
I see no reason to turn off the valve to the carbonator....that doesn't really accomplish anything....just run it as it's designed. It turns on when it needs too. If you want to bleed the warm water out of the line leading from the refer....just run the soda side of the valve untill you get cold water. I'm not sure which machine you have, but there should be a manual lever under the valve cover that you can actuate....
You can turn off the water to the carbonator if needed though. It operates on a probe/level proximity switch.....so when the water level inside the tank drops to a certain point, it turns on the pump. If the water is shut off and someone pours enough drink to activate the carbonator, it will run forever cuz it's not getting water...it can burn the motor up.....some do have a overload that will trip...but it can damage it.
If you do use ice......draining the unit is no big deal. You can just drain it into a 5 gallon bucket. The only thing that drains into it is what you pour in the drip pan and Ice melt...so you drain out how ever much you put in....you can control that.
I just realized how much I rambled. Hope this helps. Good luck.