To improve the focus of your nested tmux you may hide the status bar.

Edit ~/.tmux.conf and add the following code and restart the tmux

# Toggle status bar visibility
bind b run-shell "tmux setw -g status \$(tmux show -g -w status | grep -q off && echo on || echo off)"

Now you may use Prefix+b by default it should be ctrl+bb I did it b for bar, you may change it for your convenience.