I always find it annoying to manage nested tmux sessions so I usually use Kitty's keyboard driven tab management and tmux for server side sessions.
So, even when nested, it's always clear what will happen. One nested session (two total) is manageable. Adding another gets annoying, but the need is rare.
Here's what I use:
set -g status-left "#{?client_prefix,#[bg=colour76]^,#[bg=colour235]_} [#S]"
set -g status-right ' | #[fg=colour76]#h#[fg=default] %H%M%Z'
But yeah, when I didn't have splits and things in my emulator I used something like you did. Make a remote section visually distinct. To a lot of extent I still do, just this isn't necessary through tmux anymore
bind -n S-up \
set -g status-bg white \; \
set -g prefix None
bind -n S-down \
set -g prefix C-a \; \
set -g status-bg green