tmux sets the TMUX environment variable, so in .bashrc (or .profile or whatever):
if [ '' = "$TMUX" ] ; then
echo not in TMUX
else
echo in TMUX
fi
You can set INPUTRC to override the default .inputrc location, so you could have a tmux one and a non-tmux one, and export a suitable INPUTRC value in .bashrc depending on TMUX. You could even concoct a suitable .inputrc (e.g. in /tmp) for that session based on a "common" file and a "non-tmux session" file.
Unfortunately tmux exports TMUX, and so subshells started from a tmux session will have TMUX set regardless. Not found a way round that yet.