You wrote, "...Berthier was alone in the room when he fell out of the window,.."
That's only if you believe the housekeeper. As I recall, it took a long time to finally get her interview. Plenty of time to make sure she knew the right story to tell. Single witness is useless without corroboration. She might have been part of the plot for all we know.
People rarely fall out of windows. The window was up high; the reason for the chair. .
Berthier was widely known as N.'s right-hand man. The Russians aren't going to care one bit about his health or his capability of serving N. All they knew was that he was there within their reach, he was insisting on being allowed to return to France; Russians are not going to believe for an instance that this was for any other reason than to return into N.'s arms, just like Ney. Hard to imagine that the Russians didn't seriously consider killing Berthier. Maybe I'm just bloody minded but I know for certain if I was a Russian soldier and had the opportunity to assassinate the number 2 N. officer; send in the drone strike.
I also recall that the description of the body was gruesome with many impacts seen on his body. Something about hitting outcropping on the building side; however, a picture of the building shows a straight drop. Russian assassins would likely have wanted to make sure Berthier was dead or dying before he hit the ground. Couldn't use a knife or gun or even a garrote; blunt force trauma would be the ideal solution for this situation.