It can get quite busy up to the point where they do not accept paid entry anymore for capacity reasons. A Tuesday morning should not be nearly as bad as a Monday, but you could still face a rejection.
If the lounge is really busy, yet they still accept paid entries, personally I would not bother - it will not be a pleasant place to be! But if at reasonable levels, I would pay the money if I were you. It will be a place to freshen up and relax away from the crowds outside. I believe many of the other lounges at CPH accept paid entry at around 250 DKK too though, which might be worth looking into as an alternative.