There is an approach to use tmux inside tmux without using different set of keybindings. That's what I have been using for couple years and I really like the setup. I have the following binding in my local machine:

    bind -T root F3 \
      set prefix None \;\
      set key-table off \;\
      set status-left '#[bg=#C678DD,fg=#2C323C](pass-#S)' \;\
      set status-style bg=#E06C75 \;\
      set window-status-current-style bg=magenta,fg=black \;\
      refresh-client -S;

    bind -T off F3 \
      set -u prefix \;\
      set -u key-table \;\
      set -u status-left \;\
      set -u status-style \;\
      set -u window-status-current-style \;\
      refresh-client -S;

This way, pressing F3 in local machine disables the prefix and I can use remote tmux as if it is local. When I want to get to local tmux, I press F3 once and I am back to local tmux. I use highlighting to easily show that local prefix is disabled.