Why not just use mosh to establish & maintain the connection then then use tmux/screen to manage the terminal sessions?
mosh is a great idea and sometimes useful when you are changing connections or opening and closing a laptop all day but most days I think I am better off without it.
Also I rarely have a remote connection open without forwarding some ports about so mosh often doesn't do what I need anyway.
Unfortunately every layer of terminal emulation tends to filter out some terminal capabilities. It is a small thing but I like to be able to yank a selection out of a remote vim and paste it locally and last I saw mosh's terminal emulation eats the necessary control codes.