If you go over the bridge to Sweden there a few aeroclubs not too far away.
Malmö/Sturup(Mostly Pipers, DA40) -
Välkommen till Sturup Flygklubb
Söderslätt(Mainly UL) -
ESTT - Hem
Landskrona(Piper, Cessna) -
Landskrona Flygklubb
I would guess that these clubs are within 60-90 minutes drive from Copenhagen.
Seems as only Landskrona's site is available in "healthy" english so you can contact me if there is some information you need that you can't find.
I'm not a member of any of these clubs though I have friends that are.
Good luck!