Personally, I’m a fan of wet/dry vacs (aka shop vacs)… preferably one with a 1 3/4” or larger diameter hose (rather than 1 1/4”). Pretty much anything other than a shop vac tends to eventually succumb to the demands a bunny room places on it. With a shop vac, worst case scenario is that the hay clogs the hose (which can be easily fixed with a wire coat hanger).
I’m not sure about Denmark, but in the US you can get a decent bunny-friendly wet/dry vac for $50 or less at any home improvement store (a decent traditional vacuum costs at least $70). As an added bonus, if your bathtub ever clogs late at night and you figure you’ll deal with it in the morning only to have a god-forsaken tree roach (which, if you’re like me, you’re terrified of) pick that exact night to invade your home and suicide at the bottom of the half-full bath tub, you’ll be able to deal with it without violating your “I don’t pick up roach corpses except with a vacuum” rule, heh. (That’s actually what prompted me to go out and buy the wet/dry vac in the first place, lol… but it sure came in handy with the bunnies afterward!)