If you want to add anti-freeze, I would go in the opposite order. The thing is, the "cap" should be the lowest point, so you would have to be stuffing anti-freeze "up" into the pipe.
I would thaw, drain, then cap it. And then
pour a bit of antifreeze "down the drain" in your kitchen sink. The anti-freeze should lodge in the low point, by the cap, where your ice clot is now. That should take care of it, and no worries about foaming or off tastes in the spring from that location (assuming you don't drink out of the drain
Then just don't pour any more water into the drain until spring, so you don't dilute the trapped anti-freeze.
Or, if you are sure there are no low points (in other words, no place for water to be trapped), you could leave it as is.
PS: I'm not sure what the screen is for. Is there any chance you are confusing the inlet (where you would hook up a hose for "city" water) for the drain? That shouldn't be the case if you have a male, because the city water inlet should be female, but I'm just wondering because of the screen. Another possibility is that former owners left the cap off for the winter, thereby allowing the drain to be "open" and not vulnerable to freezing, and the screen was to keep pests out. I would think otherwise the screen would clog with the usual sink "debris."