[VariousRen]

I wouldn't impose disadvantages, because the player isn't actually out of control of their actions when they kill people.

In one of my weekly games I play a fighter-wizard who is most definitely callous (In D&D he would be solidly chaotic evil). He is not, however, Callous with a capital C, because he's perfectly capable of acting and reacting normally to people. In your case Bloodlust would be for a character who must kill people when they get the chance, your player has just shown a fondness for that particular route over the alternatives.

Instead I would look at advantages that reflect the characters particularly large experience with murder and suggest they pick that up, or inform them that it's available when it otherwise wouldn't be. Your murder-fond player might grab a level or two of fearlessness, or maybe you change the definition of scary for them with respect to fright checks.