Most cemeteries have one day per week when flowers and other objects left at gravesites are regularly removed. It's part of the maintenance since objects on the graves interfere with mowing the grass. People placing things there are usually aware that the object will probably not be there for more than a week, so I wouldn't worry too much about the missing teddy bears becoming an issue.
Someone may decide to try to get rid of the pup though if he's seen to be a nuisance. It's a good idea to reinforce the fence to keep him confined.