I knew June was going to be tough, but I guess I was overly optimistic.
The return options were not great via AA/BA even without using mileage! One BA flight left Copenhagen early in the morning, too early coming off a cruise, and the other required an overnight in Heathrow. I thought - ok, which is better: staying the night in Copenhagen or booking an airport hotel in London? and ultimately decided that it was a little ridiculous to add an unplanned night to our trip, for whatever minimal savings we'd get after factoring in the hotel and meals. SAS has multiple afternoon flights out of Copenhagen to Chicago, and we have just a 3h or so layover from there to home. They have a lounge in Chicago plus one in Copenhagen of course, so it made more sense to spend the money on the better flight options that came with no added cost/hassle afterward.