I do think the Mystery man is Bae BUT I also think that all the things in the apartment also allude to The White Rabbit. The clocks, the monocle’s, the Hatter poster…add to that the way he is dressed (a suit) similar to the White Rabbit’s attire…also the phone falling out the window as if falling down a hole (a rabbit hole)…
So lets see, Regina has been to Wonderland as has Jefferson before(I hate Wonderland) maybe just maybe and I’m just spitballing here, Bae met Jefferson in Wonderland at some point and maybe Jefferson even helped Bae escape Wonderland. (The Mad Hatter and The White Rabbit did know each other). It’s even possible Regina could have known Bae was in Wonderland and his connection to Rumple. It wouldn’t be unheard of for Jefferson to just randomly go around helping people Rumple cares about. Plus, there’s the fact that Jefferson KNEW of a connection between Belle and Rumple. How? How would he KNOW that Belle spent a short period of time with Rumple in the FTL? Unless, he was perhaps friendly with someone who had a REASON to keep tabs on Rumple….like his SON. Which leads me to believe that A. The Mystery Man in fact Bae and B. That he at some point lived in a world where having the trinkets he has is normal and that SOMEONE from SB knew him well enough to let him know the curse had been broken. So my long winded guess is he’s Bae/The White Rabbit.