its simple just take your time and clear all the guards outside, actually just inside the mansion should suffice I guess coz I just darted campbell, he ran away and just when he stepped outside the mansion I got the notification that objective is complete he is safe (did that have to do something with ghost? dies he "detect" you? coz thats the only mission where I couldn't be ghost)
That's definitely a detection, yes. Just think that, at this point, there is absolutely no reasons for him to believe Campbell is trying to kill him, so he is not really more than a regular guard/npc, reacting to an aggressive action against another npc.
You would either have to nab Campbell without the Captain noticing ( which is very hard considering campbell is always in front of the Cap' ) or very quickly dispatch them both.
As I stated, the most 'clean' way imo is to wait for them both to go in the secret chambers, wait for campbell to reveal its intentions, and take him out at that moment. This way the captain knows you are an ally and no penalties will ensure ( other than having to haul Campbell's ass back to the interrogation room if you're going the non-lethal way ).