You can make a cooler work with any welder. A lot of these small inverter machines have cooler options but they are a rip off. They add loads onto the machine price. Id buy or even make a cooler separately,works out so much cheaper. Not sure if Jassic do one but I believe Rtech might.
Can use a water cooler with any TIG, obviously need a water cooled torch too. AFAIK none, or little, of the cheapnese stuff is designed with a matching cooler* in mind although some of them have an auxilliary power outlet that'd save having to indepentantly switch on a generic cooler
* By matching i mean something that's designed to bolt to the bottom of a specific machine and be fully controlled by the welder for example my 200A Lincoln can be carried about in one hand complete with cooler and the machine switches the cooler on when you strike an arc in TIG mode and switches it off again when the machine is left idling for so many minutes. Like all the other big brands that do similar things the specific coolers come at a HEFTY premium for what's basically a pump, resevoir, heat exchanger and maybe a fan. At the time i bought mine (nearly 14 years ago) the water cooled package was an extra £5 - 600 i think. I was wincing when i got my wallet out but am glad i did... soooooo much nicer than having extra boxes to cart about to a mobile job, less things to connect/unconnect, can't forget to switch the cooler on etc.
Is the aux socket designed for a cooler.
Mine has the socket on the cooler, the mains cable from the machine plugs into that socket on the cooler. The mains cable on the cooler goes into the wall which powers both the machine and cooler. Mine is not as clever as Hotrodder's lincoln in the respect mine runs consistantly when the machine is idling.
Are you sure the aux socket is not a remote or pedal socket.