Why not just keep your key bindings compatible with screen? You can't stay with legacy systems forever.
Vertical splits were patched in and added a long time ago, screen is usually installed by default, so I know it is there on systems where I'm not allowed to install packages. I can share screen sessions with co-workers, split windows etc.
IIRC, that patch floated around for a while.
The other thing that tmux brings to the table is better scripting support than screen + things like tmuxinator (for me at least).