Are you after the most comfortable seat/most premium experience, or the shortest, best-timed flights?
Keep in mind that European airlines generally don't offer premium cabins on their intra-European flights. "Business class" usually just means a blocked middle seat and some complimentary catering. The longer the connecting flight, the more this matters. On TAP, going GIG-LIS-CPH, you'll sit on a cramped and decidedly un-premium shorthaul plane for three-and-a-half hours (they at least have decent legroom in their shorthaul business class) On LH, AF, KLM or BA this won't matter as much as FRA, AMS , CDG or LHR to CPH is a short hop of about an hour.
If you fancy widebody comfort with lie-flat seats all the way, book on Emirates where you get an A380 from Dubai to Copenhagen... but at the expense of significantly longer flights.
Avoid connecting in the USA. The US doesn't 'do' international-to-international transit, and you'll be subjected to the full immigration 'experience' and need a transit visum.