I don't have any suggestions about keeping the food from getting spilled and I don't know what your set up is, but I really like the way I've developed to deal with what would otherwise be waste. I have geese and ducks and I am able to leave the door to the rabbit area open during the day. I can close of the yard where this door opens so that only the geese and ducks can get in there- no sheep, cattle or goats, and the geese LOVE to go in and clean the floor for me. They sound like they are having a party when the find the door open. Talk, talk, talk, pick,pick, pick. The rabbits don't seem to have any problem with these visitors and I'm just happy to have animals eating up what would otherwise be "waste."
Sometimes when i don't want to leave the door open for whatever reason, I rake up all the manure, spilled feed and hay and trundle it up to our orchard to use a mulch around the trees. the geese like that too since they have access to the orchard and they will follow me up there and begin picking through my pile, sometimes before I've even finished spreading.