I've finally gotten around to sorting out running nested tmux instances. I found the base for the configuration in the article Tmux in practice: local and nested remote tmux sessions, which links a few other related resources.
What I ended up with was this:
# Toggle tmux keybindings on/off, for use with inner tmux # https://is.gd/slxE45 bind -T root F12 \ set prefix None \;\ set key-table off \;\ set status-left "#[fg=black,bg=blue,bold] OFF " \;\ refresh-client -S bind -T off F12 \ set -u prefix \;\ set -u key-table \;\ set -u status-left \;\ refresh-client -S
It's slightly simpler than what's in the article above, but it works and it fits rather nicely with the nord theme.