Oh crap, I don't think you're going to be able to do it.
One possibilit I see, and I don't know if wifey would, or if it would fit, would be to cut the hose that the nozzel is on and somehow fit a set of small quick connects to the ends, and modify you IC to have one too.
Something like this, only smaller (I don't know if they exist).
Oh, Oh, Oh, There's another option though. Back in the day I used to do freelance photography, and my first darkroom was just literally a bathroom where I couldn't do anything fancy with the plumbimg. So I rigged everything I needed to hook up, like film real rinsers and such with those cheap hair washing things you put on a sink that doesn't have a rinser. You can usually get them at dollar stores.
I just cut the sprayer off the hose, and built everything with nylon barb fittings like these.
Then I could just swap out one thing for another, leaving the hose connected.
You could affix this modified hose to your chiller really easily with the same metal clamp already holding your hose. Just og to the dollar store or even hardware store and get one of those hair washers and see if it fits over your existing faucet end. You might need to use a zip tie or something to help secure it during use, but I'm sure you could come up with something.
That might be more advisable than cutting your faucet hose.