The sticky tape idea is actually not bad in theory--hey, you can try a trick people use to prevent dogs from jumping on the couch. Basically you cut up a... those prickly mats people clean snow/mud/dirt from their shoes in strips, and hot glue the strips to the tub cover. So it prickles their behind when they try to sit/lay on it.
The thing with cats is their claws/saliva have special bacteria that's toxic to birds--so even if you "save" a bird from a cat, without proper antibiotics, it'll still die. I don't know how dangerous it is to other animals but I don't know if I'd risk it.