Thanks, I tried. It still showed the same error once or twice, but somehow I suceeded to install.
I added
/usr/share/lazarus/4.0.0/components/opengl/
in
Main menu — Project — Project Options... — Compiler Options Paths — Other unit files (-Fu):,
which still does not seem to make sense to me, since I expected to install the component for the IDE, not for the project 
Hum, strange, you should install only
bgrabitmappack.lpk, nothing more and all should be out-of-the-box.
By the way the window manager Cinnamon is buggy ("by default" they say ?¿!!), it does not follow the X11 atoms rules, but this is a other story.
I have installed a VM of Linux Mint 22.1 Cinnamon 64 bit, + Lazarus 4.1 + BGRABitmap package, all seems ok, could you give a demo (or link to) where you get problems?
Picture of demo /bgrabitmap/test/test4lcl