I agree that bounces should be minimized. Still the most logical solution would be that the MTA asks the MDA about the quota status.
Yes, that could be doable. Although, dovecot's LDA doesn't actually support this. And then there's the issue of race conditions: You ask it, accept the mail, and when you go to deliver it the user has saved gotten another mail in the meantime. At that point, you probably should just accept it anyway, but if the LDA checks the quota again you'll end up with a bounce message...
However I don't know any way to let postfix ask the MDA about quota status. Even bypassing the MDA, do you know of any software that plugs into postfix that can check filesystem quota's?
Sorry, I know nothing about postfix at all. And even for exim (which I use) I couldn't tell you off-hand, but it's most likely possible to do.
johannes