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Too many factors you haven't mentioned.
Where does the pergo end (in relationship to the living room side of the jamb)??
What is the height differential??
Normally, - - the transition between floors should be below the door itself (when closed), - - so you're only seeing 'one floor' from either side.
If the pergo ends right along the living-room-side of the jamb I would either make a wide oak (or whatever matches) saddle, - - or, - - beings it's a bathroom, - - install a (4" wide) marble saddle. The marble saddle would butt about 1/8" shy of the pergo, - - leaving room for caulk (and expansion).
If you'd prefer a custom (you'd have to make it) 'over-lapping' oak saddle, - - then yes, - - let the tile fall at (the very) least 1" short so you can make that part thicker and allow for pre-drilling and attachment screws.
In the case of the oak (or similar) saddle, - - you would rabbett out each side separately to allow for any height differential, - - in the case of the marble saddle, - - you could probably find a 'compromise-height' that would allow both sides to 'butt'.
The last choice (IMO) would be to buy the 'pre-fab' transitions, - - but I doubt they would have anything that would satisfy yours (or the customer's) need, in this instance.
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