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'm sorry I wasn't clear. This would be a binding in your local tmux configuration that sends the prefix command to whatever is in your current pane, allowing you to have a key act as a 'remote prefix' button.
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.
For example, to bind C-a to control the remote session: