Henry's character is John Lark.
He might have lured Ethan into impersonating Lark because he knew that Lark's life might be in danger in that scene.
Now about the question,
- The person who got killed in the washroom might be another guy impersonating as John Lark to meet White Widow and get hold of the plutonium plates & the person who wanted them.
- Remember, August hit that guy's head hard with the laptop that was to be used for creating the mask. Also, in one scene we see August getting a broken phone from this guy's pocket. But when he hands a phone to the CIA head it's clean as a new one.
- This all points sum up to August framing Ethan as John Lark to avoid from any threats that he might face.
- The guys sent to kill them in the club were supposedly from Apostle (or what's left of Syndicate) to finish John Lark & White Window and get the plutonium plates as well.
Ref - http://missionimpossible.wikia.com/wiki/White_Widow
"Unable to create a mask of the man, Ethan assumes the Lark identity
and meets with Alanna, hoping that they've never met in person. This
proves to be true, but Ethan and Alanna learn that members of the
Apostles, the remnants of the rogue Syndicate, are sent to kill them
both and steal the plutonium. Ethan helps Alanna escape and continues
impersonating Lark to acquire the plutonium. She provides a single
core in good faith, but says her clients require the extraction of a
political prisoner as the price for the two remaining cores. The
prisoner is revealed to be Solomon Lane, the former MI6 agent who
formed the Syndicate and whom Ethan and his IMF team captured two
All of this was done to kill any person knowing of the entire deal, the dealer and getting the plutonium plates as well. Another article refers to that entire meeting being a hoax and main motive being to free Lane,
ref. - https://screenrant.com/mission-impossible-6-villains-explained-henry-cavill/
"When the movie begins, it seems that John Lark has merely hired the
Apostles to get their hands on some plutonium so that he can carry out
some mass destruction. However, it turns out that the exchange is just
a ruse to free Lane, and that both Walker and Lane are working towards
the same goal - albeit with slightly different priorities."