Danny Arce claimed he ate lunch with Dougherty then went outside with the others. Who knows where Dougherty went after Arce left.
As for the problems......
Maybe Dougherty left the domino room after eating with Arce. We know that Norman and Jarman left the front of the TSBD about 12.20. Arce had seen Jarman out there before he left.
There was minimal construction of the SN required. The stacks of books had been moved there to allow for the floor laying. Similarly a rifle could be assembled at any time that morning as long as the assembler could be assured of privacy for the required time.
Rowland saw the man with the rifle at 12.15. Williams only vacates the sixth floor after Jarman and Norman arrive below him. Has to be around 12.25.
Rowland testified he saw an African American in the SN window also. Just so happens Williams lunch remnants were original found in the SN. A piece of unfinished chicken and a bag found separately.
I wonder who that African American Rowland saw could be. What really made him decide to venture downstairs?
In this transcript of Dallas Police tapes at 12:22 the motorcade is "approaching Main".
Somewhere between 12:22 and 12:25 the motorcade is on Main.
Assume a minute or two for news to get through to those at the TSBD that the motorcade is on Main.
This scenario has Norman and Jarman deciding to go up to the 5th around 12:23 at the earliest.
Jarman seems to think they get to the elevators between 12:25 and 12:28 which seems to accord with the transcript.
This has Williams eating his lunch in the Sniper's Nest until 12:25 if not later.
Unless another black employee joined him it's hard to get away from Williams being the man Rowland saw in the SN window.
It's hard to imagine the assassin just hanging around hoping Williams would disappear.
I imagine Williams was told to get out of there and keep his mouth shut or he knew the assassin was there all along and it wasn't a problem.