Do I need to invite Sylvain to complete Share a Bite?
You do not have to invite Sylvain to complete Share a Bite. In fact, there wasn't even an option to invite Sylvain when I went to the dining hall. The Quest Completion dialogue seems to imply you can't invite Sylvain to your first meal.
Heh, so you're getting right to it! I'm impressed to find someone who's more proactive than I am. Nice going, Professor. All right, then. Next time will be my turn!