Admittedly, Six's morals are pretty damn wobbly, but they're often a case of justifying the ends. So it's not so much a matter of how far she's willing to go (because... yeah... she's absolutely equal to the task of making that 'welcome' message into something from the Grand Guignol), but whether or not she's got it into her thick skull that Negan is a tyrant worth keeping around. Supporting tyrants is kind of her special ability.
She'll think he's a psychotic dick, but she's gotta admire what he's done with the Sanctuary (THEY HAVE A LIBRARY), and she'll not miss the fact that -- psychotic dickiness aside -- he's got his own (heavily distorted and seriously effed up) moral code. Six gets that people sometimes need despotic assholes to keep their worst natures in check, so she can get behind that. To a point.
(Probably right up until she starts thinking she might do a better job running things. *coughs*)
But either way, absolutely! Six'll be happy to run messages back and forth, even the non-violent ones.