As a character
The main problem is the existence of the movie myth that suppressors (often called "silencers") can kill all of the sound of a gun... in fact, they do not "silence" a gunshot. It "suppresses" the sound, meaning it reduces the sound but does not make it go away completely.
An unsilenced gunshot is around 140 to 160 decibels -- that's in the range where hearing it once can permanently damage your ears. If you've never had a gun go off next to you, trust us when we say it's loud enough that your whole body will flinch at the sound of it. A silencer can get that all the way down to 120 or 130 decibels, aka the sound of a jackhammer. Still loud enough to cause physical pain if it's close enough to you.
So a silencer really just makes a large gun sound like a smaller gun. If you're James Bond and are sneaking into the enemy's compound with a silenced pistol, you're basically hoping the guards will decide your gun is too small and wimpy to be a serious threat, and leave you be.
Even a handgun will make noise when suppressed, and Ilsa's sniper rifle would certainly make noise.
Here's a video of different types of guns being shot with and without supressors:
Pay attention to the echo.
And this guy's outside, while Ilsa is in a room designed to acoustically project the sound from the stage out into the audience.
Here's another video of a suppressed sniper rifle. Despite being suppressed, he's still wearing ear protection:
So, outside the suppressor issue, remember that she's on a planned operation and she's trying to stay hidden. If the plan for the operation requires her to shoot him at that specific time, that's what she's going to do so - particularly considering the backup measures Lane put in place should she not follow through. If she was told by Lane to shoot at that specific moment and she didn't, that would be the other people's signal to do their jobs.
As a Viewer
What makes movies, particularly movies in the action genre interesting is the use of suspense. It's been a major part of movie making and was, in particular, a point of interest for many famous directors, particularly Alfred Hitchcock, the "Master of Suspense".
There is a famous quote of his about the difference between suspense and surprise:
“There is a distinct difference between "suspense" and "surprise," and yet many pictures continually confuse the two. I'll explain what I mean.
We are now having a very innocent little chat. Let's suppose that there is a bomb underneath this table between us. Nothing happens, and then all of a sudden, "Boom!" There is an explosion. The public is surprised, but prior to this surprise, it has seen an absolutely ordinary scene, of no special consequence. Now, let us take a suspense situation. The bomb is underneath the table and the public knows it, probably because they have seen the anarchist place it there. The public is aware the bomb is going to explode at one o'clock and there is a clock in the decor. The public can see that it is a quarter to one. In these conditions, the same innocuous conversation becomes fascinating because the public is participating in the scene. The audience is longing to warn the characters on the screen: "You shouldn't be talking about such trivial matters. There is a bomb beneath you and it is about to explode!"
In the first case we have given the public fifteen seconds of surprise at the moment of the explosion. In the second we have provided them with fifteen minutes of suspense. The conclusion is that whenever possible the public must be informed. Except when the surprise is a twist, that is, when the unexpected ending is, in itself, the highlight of the story.”
In the case of *Mission Impossible film, the scene is a very obvious homage to Hitchcock's The Man Who Knew Too Much. From a commentary piece I found very interesting in the New Yorker:
The scene pays deft homage, as Anthony Lane notes in his review, to one of the great musical setpieces in film: the climax of Alfred Hitchcock’s second version of The Man Who Knew Too Much, in which Bernard Herrmann conducts the London Symphony in Arthur Benjamin’s “Storm Clouds” Cantata at Royal Albert Hall while Jimmy Stewart hunts for an assassin. Hitchcock generates extra suspense by giving viewers a glimpse of the score: when we see a passage marked “poco a poco crescendo,” we sense that the attempted assassination will take place at crescendo’s end. Likewise, McQuarrie shows one of the killers flipping ahead in a score, to the climax of “Nessun Dorma”: Calaf’s final cry of “Vincerò,” with its rise to a high B and step down to an A. The last note is helpfully circled in red. You don’t need to know how to read notation to realize where this is headed: the high note pulls the trigger. Performances of “Turandot” are bedevilled by the problem of the ending, which Puccini tragically did not live to write. Here, the police decide the issue by removing the singers from the stage.
Much as Hitchcock built suspense in his films, the director of Mission Impossible - Rogue Nation, Christopher McQuarrie built suspense by giving Ethan Hunt a deadline that he isn't aware of. The audience is held in suspense, wondering if he will make it in time to prevent the assassination attempt with this obvious deadline looming.