Actually that is a fairly good idea, it's a whole lot cheaper than buying a compressor size 2-3 times bigger than needed. More tank volume helps, but do you have room for a 150 gallon tank? I don't.
How close is your house AC unit? What are the chances that you could reroute your AC lines to a cooling box nearby/next to etc, and simply run the air hose first to the cooling box, and then to your separator, than to the tools. You would need to remake two AC lines to lead the coolant outside to the cooling box, and back.
I thought of that because that's what I'm going to do for a cooled driver's seat. I'm going to have two new AC lines made, to lead to and cool down a box full of water. You can buy an evaporator/condensor and mount inside of the box. Just a thought, my project will cost about $250-300 I think, including a $150 cooling seat cover.