It's been too long for me, too, so I called up the scene on youtube. There doesn't seem to be an in-movie answer that satisfies, but I would suggest that it's possible Ethan's glasses are activated and begin broadcasting only when he opens them? As soon as the broadcast starts (a couple seconds before he slides them on his face and provides Kittredge with the image of Phelps), maybe the watch beeps, prompting Kittridge to check it?
Yeah- really the only part I cringe at because Phelps just stands there and does absolutely nothing but look shocked. He knew what those glasses were and didnt do anything?