Earlier quoted context omitted.
You can configure your local tmux to have a key press send the prefix command to a remote tmux with a single keypress. For example, to bind C-a to control the remote session: bind-key -n C-a send-prefix
Binding isn't an option as only defaults are available on target machines (can't save .screenrc file). There can be thousands of target machines so binding doesn't scale.
I agree that bindings still don't scale incredibly well depending on the number of host machines you need to ssh from. I think your screen workaround is a pretty neat solution.