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.
Serious Eats has a very comprehensive guide [1]. The short version is rub lightly with canola oil, put in hot oven for about 30 minutes. Take it out and oil again. Repeat 3-5 times.
The only thing you need to avoid while cooking is overly acidic foods. If you try to make a tomato sauce in it you will see the seasoning flake off. That's not good, but is very easily fixed by repeating the above steps.
To clean it, scour the plan with kosher salt and a paper towel. Once clean, coat lightly in canola oil.