How do I transfer ownership of a house when completing a quest?

You can't change Ownership in-game. It has to be done in the CS as a mod.

Actually, MWSE might be able to do this, (just a guess) but it would still require you to set it all up in the CS anyways.

I know it has to be done in CS but what I mean is:

1. Do quest
2. Get key to house from previous house owner, the ownership needs to be changed so the PC doesn't get a bounty

What I mean is... You can't change Ownership by scripting or dialog, so that can't be done as part of your Quest.

Either have no Ownership to begin with, so when the house is yours the containers are already safe.
Or, jump into the CS when you're done the quest and remove the Ownership.

Or make a new cell, copy/paste everything from the old cell into it, remove ownership on the new cell, and make a scripted door that changes from one version of the house to the other. The cell can't have the same name though, so it would go from "so and so's house" to "something else manor" or whatever.

If you don't want to change the name of the cell you can make your duplicate copy in the same cell, just place it far enough away that it won't show up on the mini map. You can avoid the bugs associated with scripted doors by giving each copy of the entry door (one for the original cell and one for the unowned cell) a unique name. Place an invisible activator and make a script on that activator to disable/enable the doors.

theoretically, you could have a copy in the exact came location except the copy is directly over top of the other, so there would be no change in the map (cuz its a top down view of the map)

ST