If Bushwick were to escape completely, it would be a disgrace on Commander Isaac and the church of justice as a whole.
Commander Isaac himself would also surely receive more punishment and disappointment from the cardinals than if he were to kill Bushwick. So even if killing Bushwick now is a bad idea, it is a good idea since it will prevent an even worse situation from happening.
This is the logical reason for killing Bushwick, and it will be the one he will tell the cardinals when he is questioned for killing Bushwick.
As for the irrational reason for killing Bushwick, it is because he is very angry.
Maybe he would have spared Bushwick's life if he wasn't so angry. It is possible for that to happen. But it can't happen anymore because he is too angry.
The main source of his anger doesn't come from Bushwick or from the reports of bad news that his captains just gave him. His main source of anger is the news he got earlier that bloody hearts have been stolen again.
