Yep, that's it. Looks like Firestik makes a part that fits the bill, so maybe it's a solved problem after all.

The exact hanger I have is this one.

It would be nice to be able to hang an HT from the same bracket, like my ID-51A. Of course, that would be a different part so that it could screw in (or clip in?) to the back of the radio, but the button interface ought to be the same.