I think the stranger is not Bae, but the Tinman from OZ.
Remember, the sequence opened with a horse. One of the OZ books is called “The wishing horse of Oz”.
L. Frank Baum tried to end the series by saying he lost communication with the land of Oz. Then, we he went back to it, he claimed he had re-established communication with the help of wireless telegraph. This would tie in with all the communication equipment in the room.
The shoes hanging by the windows represent the ruby slippers. Add to that the red poster near by (rubies are red) and the poster of a human heart.
Furthermore, L. Frank Baum wrote a number of books under a female pseudonym. That could explain the female (rather than male) bowling trophy. And before he became a writer, he owned a store called “Baum’s Bazaar” – the room certainly looks like a bazaar.
And, if he is Henry’s father, it would explain why he left Emma when she was pregnant – he has no heart.