Just for context the below event type

state change

worked for me.

With my event bridge having the below triggering my lambda:

  "detail": {
    "type": ["REPLICATION_TASK"],
    "category": ["StateChange"],
    "eventId": ["DMS-EVENT-0079"],
    "detailMessage": ["Stop Reason FULL_LOAD_ONLY_FINISHED"]
  },

I apologize, i don't have enough reputation to upvote the previous individuals. Wanted to share my findings.