I ended up with a partial solution to the problem by installing `xbindkeys` and setting up the following `.xbindkeysrc` in my home dir:
"xclip -selection primary -o | xclip -selection clipboard -i"
mod4+c
"sh -c 'xclip -selection clipboard -o | xvkbd -xsendevent -file - 2>/dev/null'"
mod4+v
These shortcuts allow me to copy text into the X11 clipboard from a terminal emulator like urxvt, as well as paste from the X11 clipboard into the terminal emulator. Qt applications work for copying+pasting as well.
However, when using Gtk applications (like Firefox, gedit, etc.) if I tap the Mod4+V to paste, it doesn't seem to work. If I hold the keys down for a few seconds and then release, it pastes fine. Bizarre.