If they had wanted the attempt known, they'd have made it very public instead of hiding the Klingon away in a place obviously security never thought to find, that was only discovered after Kirk and crew busted the secret plot. Otherwise he could have gotten away rather easily and even blended in with the Klingon crowd during the panic.
There would have been no way to hide the shooting location: unlike bullets, phaser beams immediately reveal the location of the shooter!
Yet the killer mustn't be caught or it will be revealed that he's not a Klingon for real. So shooting from a balcony is the ideal choice: the victim dies, everybody turns to look at the assassin, and sees his face before he makes his well-planned escape like any realistic assassin would.
Whether the plan further involved Cartwright's cohorts vaporizing Colonel West to eliminate all evidence, we don't know. It doesn't sound likely that the Klingons played much of a role in the attempt - sure, Chang is on orbit, but he's on an independent hunt for Kirk and need not even be aware of what is happening down on Khitomer!
You'd think they would have targetting systems, or even some fanciful tech that can pick out and display a target without sighting it.
Sights are useful for studying the victim, though. Targeting is worthless if the target can't be identified and otherwise assessed.
Timo Saloniemi