Ok so you're not really looking for a traditional stay-on-top feature, as what you are talking about is a pop-up menu, never meant to stay on screen..
What you seem to be asking for is something that would be mostly functionally equivelant, and that HAS been asked for in the past, a kind of tiny window with recent (quick paste) items that you can (double) click to paste into last active window.
I note that the main window will do exactly what you want (and has a stay-on-top option in view menu), it's just that it's a bit big to keep on screen at all times, correct?