I'm going to be the one to point out that, in terms of effectiveness, the stun setting is actually ideal in this kind of situation. For one thing, there's the strong possibility that the Klingon ship has sensors such that firing on a higher setting will set off a security alarm or something (see "Undiscovered Country"). This occurred to me last night when I noticed an alarm sounded in the background (for a few seconds) ONLY once the Klingons actually fired back, and yet when Burnham uses her phaser on the bridge, there's no alarm, just one really startled Klingon who yells "intruder alert!"
But beyond that, there's the fact that the stun setting can mission-kill a Klingon from even a glancing blow. The higher disruptor setting requires prolonged contact with the phaser beam to vaporize them while a short pulse will only blast a chunk out of whatever part of them you hit. Between Klingon armor and redundant organs, that would have the effect of wounding them but NOT rendering them unconscious and possibly even failing to immobilize them. So while the stun setting might leave you with a snoring Klingon on a 60 minute wakeup timer, the kill setting might leave you with a pissed off howling Klingon with a weapon still in his hand.
Switching to "kill" on T'Kuvma makes sense in this case because the Captain has a knife in her chest, his back is to her, and he fucking deserves it at this point. Considering this one lapse of judgement pretty much landed her in the slammer for life, it's not a mistake she's likely to repeat in the future.