This is really easy to remedy: just bind send-prefix to C-a C-a. That way you'll be able to easily go to line beginning inside tmux, and outside you won't type the unnecessary 'a' character.
Moreover, I forgot if it's the default or if I set it somehow, but in Emacs a single C-a goes back to the beginning of a line, but C-a C-a goes back to the beginning of text on the line, which is almost always what I want anyway. So I press C-a twice anyway, everywhere, and I don't have the problem you described at all :)