Here's the best solution I could develop (valid for Linux only):
Install AutoKey, add a script there:
from Xlib import display
c = display.Display().screen().root.query_pointer()._data
x = c["root_x"]
y = c["root_y"]
mouse.click_absolute(x,y,3)
keyboard.send_key("<down>", repeat=10)
keyboard.press_key("<enter>")
keyboard.release_key("<enter>")
This script makes AutoKey press mouse-3, and then it presses ArrowDown 10 times (I have a 'Copy Image' there), and finally it presses Enter -- so that pic under Your current mouse position goes to clipboard. You can then bind this script to a key in AutoKey. Works more-or-less stable (don't call it several times at ones).