Your turn ends if you start a battle by playing a Battle Tile. Does starting a battle by filling the board also end your turn?
Yes, it does. A battle will always end the turn. From the 3.0 rules:
The player who starts a Battle (by playing Battle tile or filling up the board) finishes his turn, regardless of the number of tiles he has left. As soon as the Battle commences, no other tiles can be used or even discarded.
It's also the way it is implemented in the mobile app (I've checked, forcing the described situation).