Will I have a code problem attaching a 1.5“ grease interceptor to a 4” drain pipe

I have had my Bosch dishwasher for about a year using the same setup you contemplate, so I'm familiar with the exact drain connector you have on the dishwasher hose. On mine, there was something of a taper between the 1/2 and 3/4 section and being rather flexible rubber, and I had no difficulty stretching the 1/2 section of the fitting to fit on the 5/8 gap fitting. I then tightened down the hose clamp with a hex bit on my cordless driver. The clamp was tight enough to somewhat compress a groove into the rubber, and I tugged on the thing as hard as I could. It isn't coming off IMHO. I'm at work, but can send a picture later if you need.

I understand you are worried somewhat about the quality of the connection. You could get a 1/2 to 5/8 adapter like this:

But that would entail 3 hose clamps and a section of 5/8 ID hose to connect between the adapter and air gap. That seems like more points of failure. At any rate, I think you already have what you need to get this working.

You can use something like the non-toxic RV antifreeze which evaporates more slowly than water; Or wash the basement floor occasionally.

A touch of mineral oil (the stuff sold for putting in people) may help to prevent evaporation by forming a surface film, but don't overdo that.