You can configure tmux to synchronize with the X clipboard, although it is somewhat fiddly because of how terminal emulators work. See this page: https://github.com/tmux/tmux/wiki/Clipboard

In most terminal emulators you can also use Shift (or sometimes a different modifier) to bypass application mouse mode and have the terminal emulator handle the mouse event. Of course you can't copy out of the history with this, but that is another limitation of terminal emulators.