You should not read anything into the "Business" class designation. It's just there because the flight carries a J/C/D/I/Z fare class and certain DL systems will call it "business" based solely on the booking class. It's not a "business" class flight in the sense of D1 service. Note that the delta.com booking search actually refers to the seat as "First" when you do a flight search (as it is looking at more than that just the fare class to label the flight).

You will get lounge access because you are in a premium cabin on an international flight. There's an exclusion for flights to the Caribbean, but since CUN is in Mexico, it's not excluded.