I have this problem at both places where I work, but with gmail. In both companies, the gmail account was registered under the business name (=paperwork name), but the trading name (=public name) is different. So you have to train your users to log in with the business name, even though that's the only time it's ever seen by anyone except the lawyers.

Then you have the problem of emails going out under that name instead of the trading name - while the aliases generally work well, you have to train users in setting their accounts properly. I've contacted gmail support about switching over names, but their response was "it'll take around three days, and you may not be able to access email for that time", sprinkled with a tone of "it'll probably work without a hitch".

Yeah, you try telling the business side staff that they might not have emails for three days in a busy place... unfortunately I was too spooked to proceed, so can't vouch that it works :)