I assume MC will send Charlotte to take Emma home, then call his bartender warbuddy to come help him dispose of the body.
Sidenote: You know you have a true blue ride-or-die friend if they'll drop everything and come help you hide a corpse
MC knows enough by now to know...or at least suspect that Carter is involved with the Sex Slavery Ring, and might reasonably anticipate that if he tries to tell the truth he'll just get setup by other corrupt cops.
Also judging from bits of slang sprinkled throughout this game I believe it's all set in Europe. Britain, most likely. And Europeans don't have nearly the level of respect for self-defense that Americans have. Red/Purple State Americans, at least. So while MC was morally right to kill Carter, it's possible he might still have violated some sort of duty-to-retreat law. A psychotically unjust law, but a law nonetheless. Further incentive to hide the body.
Maybe the agent lady will even help.