Why was Alfred OK with Bruce going out to murder people?

I think there's a difference between his being OK with it, and his realizing that nothing he says/does will stop Bruce though at least maybe this way he can either bring Bruce back to his senses faster or he can keep Bruce alive long enough for Bruce to come to his senses on his own. I mean, we see him poking at Bruce about it in the beginning of BvS. He shows his clear disapproval of it, yet he knows he can't stop Bruce. On one level, it could be that he's the butler and Bruce is the master, so he can express an opinion but can't dictate to his employer. Though I don't think that gives Alfred enough credit, I think this is the last person he has left (we've had no mention of his having any other family, etc.), and he worries what would happen to Bruce if he were to leave him to his Batman thing without Alfred there to try to look after him and steer him away from the darkest recesses of his soul.

In theory, this Alfred has watched Bruce lose his parents, grow up in a shining city that is crumbling in the wake of the same loss that changed Bruce's life, becoming Batman, possibly having some love (Selina Kyle, Vicki Vale, etc.), adopting a son, then losing definitely the son (hence the uniform in the glass), possibly losing whoever the woman in his life had been, the Joker/aka killer of Bruce's son getting away with all of it, and the world seemingly plunging further and further into darkness despite 20+ years of Bruce's fighting against the tide. Then, an alien is outed as being on Earth and a group of alien warriors threaten all of Earth if this guy isn't handed over, and in the scuffle, Bruce loses someone who had been a remaining link to his parents as well as dozens of Wayne Enterprises employees, and what all happened in the debris cloud where Bruce is dealing with the guy who had the beam fall on him and the orphan girl. I think Alfred worries Bruce won't be able to pull himself back into the light and that his grief will consume him completely. So Alfred begrudgingly sticks with his charge.

He may also feel some guilt about this. He didn't stop Bruce from being Batman in the first place- he let him take on vigilantism. He didn't make Bruce stop when Bruce became a father, he then let Bruce train (and may have helped to train) the kid. He, like Bruce, didn't stop the kid from taking on Joker. And he didn't make Bruce stop, and/or leave Gotham, after Bruce's son died. So I wonder if there isn't a little guilt going on there too. I mean, why let Bruce Wayne have all the unhealthy coping mechanisms?