In the days when Binion's Horseshoe had a faro bank, it would have been before they acquired The Mint next door with its hotel tower.
At that time, the Horseshoe was only 2 stories tall and had rooms on the 2nd floor only -- probably 50 rooms or less. I even stayed in one of those rooms on one trip. So #3699 doesn't make sense as a room number.

Instead, it might be a number for a guest safety deposit box, with a key attached that the guest took with them. (Just a guess though.)