If that was my "wife" (or husband in my case) you bet your hiney i'd be in that ambulance.
Possible reasons for HKS not going in ambulance:
1) Only one person allowed to ride along. Moe may have insisted he go, he seemed very protective of ANS.
2) HKS may have known (or been convinced) that ANS was not revivable and his priority would be to stay at the room and present a suicide history or something to the police in order to attempt to influence the investigation.
3) HKS may have wanted to attempt to "control" any evidence in the room.
Probably other reasons also, just speculating. I see more reasons why HKS would prefer to stay in the hotel room than go in the ambulance. Cold-hearted, yes. That's HKS.