I recently returned to playing Skyrim and started a Thief/Assassin playthrough. I don't kill anyone of the playable races for whom I don't have a contract/bounty/quest to kill. Even if I'm being being attacked I run away, hide, and sneak back.

I decided to do the Light Out quest in Solitude, which I didn't do on my first playthrough. When I reached Deeja at the Icerunner, I picked her pocket before speaking to her and realized I was betrayed - but the quest wouldn't update. I tried to go to the Broken-Oar Grotto, but Jaree-Ra wasn't there.

I reloaded to an earlier save and decided to go along with the quest. When I got to Jaree-Ra, while stealthed (97 Sneak), he automatically began talking to me (even when I used an invisibility potion), making everyone aware of my presence.

This quest, which seemed like the sort of quest for a Thief-type character, is very unfriendly to stealthy players and forces needless bloodshed. Is there any other way of completing it that I haven't managed to find? UESP and Skyrim Wiki aren't helpful on this matter.