This is what I do, I can't speak for Mint's screenshot tool, but Spectacle for KDE will indeed freeze the whole screen after a set timer - allowing you to open context menus and whatnot. Then on the "frozen" image you can highlight only a specific section of the screen to screenshot, make annotations, etc.
Spectacle is one of the things I miss the most every time I try out GNOME again for a bit.
With the MATE screenshot tool, you can set a timer and set it to either capture the active window or the whole desktop. It will capture context menus when using the timer.
Why did this not come to my head. My idea was to use video capture card, connect that to another computer or phone, set screens to mirror and then take snapshot of the video.
Testing on my own computer, one workaround appears to be to use unmodified PrintScreen, leaving a hand free for the mouse, and quickly right-click for the context menu after the keypress but before the Save pop-up appears.
A PITA to be sure, but it does capture the context menu.
As for cropping down a full-screen capture, I tend to use PhotoFlare for jobs like that (find it in Software Manager) assuming you haven't anything else installed that does the job.