I have my cages suspended from the ceiling of my
chicken coop to reduce wasted space so I never had this issue but your best bet kill the ants, if your not against poisoning them they are most likely coming from out of the ground from the hive I would locate this spot and put some of those gel traps. If you clean the box it
should be fine I mean wild
rabbits most likely deal with ants a bit being they burrow into the ground to nest, I am not sure as to what cinnamon is to rabbits but I used to sprinkle cinnamon around my bee box as it was a deterrent to most insect also borax the laundry
soap works great as a barrier as the coarse soap actually cuts the exoskeleton of insects even cockroaches, its so good at keeping bugs away it is used in waterfowl mounts as a natural preserve and insect repellant.
I clean the nesting boxes out atleast once after the babies are born not wash it down but remove old
hay and any droppings just to ensure no kits get any infections and have never seemed to have any trouble with the mothers as domesticated rabbits don't mind human scent unlike wild rabbits which is why we can handle the kits without the mother rejecting them. Also I only ever put my nest boxes in on day 28 of the pregnancy to ensure mom doesn't get use to using it as a bathroom.
Something else that may work give the cage or hut a nice little rinse down as ants leave behind chemical scent trails to find the way to food and home cleaning the area my wash away the bread crumb trail they are following to the box.