Many people aren't going to go for a funny animal Jesus. The Bible says He made MAN in His image.
I think the idea of a cat- or dog-faced Jesus is offensive. Or making Jesus any other type of humanoid animal.
I've seen a Looney Tunes parody of the Last Supper, which featured Bugs Bunny as Jesus, Daffy Duck as John and Elmer Fudd as Judas, among other characters and found it deplorable. A lot of other folks found that offensive.
Unfortunately, there's also about 50 other artist parodies of the Last Supper.