There are 2 entrances into the battle tower, where one is being blocked and has a teleport shrine just nearby. If you had activated that teleport shrine before the fight with laiken, you can easily teleport outside of your battle tower once you successfully obtain it. Then you can proceed on with your quest.
Eventhough majority area of the sentinel island is cover in poison shroud, you can still land and complete your achievement on the raft, as well as getting the magic book for the quest. Those areas are not affected by the poison.
