I have tried to configure bash for better use it (including launching it automatically) but then gave up due to not enough gain. But I never saw a need to configure tmux itself.
I can't say the same about screen. At a minimum, there's an inherent incompatibility between it and emacs.
--------------------------------------------------------------------------------
## Control-^ (usually Control-Shift-6) is traditionally the only key chord not used by emacs
escape ^^^^
## disable all the 1970s era "flow control" operations so I can actually use emacs
defflow off
--------------------------------------------------------------------------------
The first line sets the prefix key to C-M-6 which is not used by emacs. The second turns off the "ctl-f" flow control stuff so emacs gets that chord. I then use emacs itself for all "window" management and terminal multiplexing.
I run urxvt.
I also have custom fixes for rxvt that I've built up over the years, both a custom terminfo file, and fixes for emacs' terminal mapping.
I'm willing to bet your "inherent incompatibility" is a terminfo or similar terminal code interpretation problem.