Should I tile under bathroom door?
If you are planing on adding a threshold (also called a transition strip) and you are sure it will be wide enough to cover the gap between the laminate and the tile, then no, there is no reason to add the tile in that gap.

As you noted with the tile there nailing the transition strip is nearly impossible. (Could only be nailed on the hall side plus the door frame edges, you'd be mainly relining on glue.)

If you are worried about the empty space (some transition strips are way too thin to bridge a gap like that) then just put plywood in the gap between the tile and the laminate. That makes the tile job easy and the threshold piece secure.

To determine the thickness of the needed plywood measure the height of the finished tile and pick something close or a bit under, not over (in your case sounds like ½ inch will work.)