If your 6am flight is leaving the EU Schengen area directly, be aware that CPH Airport recommend arriving 3 hours before the flight - 2 hours if within Europe.
Your current plan sounds like the least stressful and most convenient to me.
It would be quite easy to take the metro to Kongens Nytorv station or the train to Copenhagen Central (København H). Personally I would really like to wake up just a short walk from the airport at that time of the "day".
The public transportation system in Copenhagen is very easy to use (disclaimer: I'm a Copenhagener).
I would also recommend the City Pass. It includes trains, metro and bus.
If you buy it using the DOT app, just make sure you have enough power on your phone for the day.
And maybe bookmark
https://www.rejseplanen.dk/webapp/?language=en_EN - the public transportation journey planner.