I couldn't kill Lilly with any of the rabbits I have so used a Pygmy Cow. Stampede and Udder Destruction works every time. However the snails still proved troublesome as well but after a few tries with bad RNG I finally got it along with my Family Battler achieve!
Use S/S rabbit, and use nether roach 1,x,x as third pet, nether roach spam nether blast, it is still a bit of RNG depends if ticker is faster than rapana whelk, as ticker without flying speed buff is same speed as rapana
Worked with a fast rabbit, I used Spring Rabbit as didn't have a levelled Noblegarden Rabbit. With a slower rabbit Lilly killed my first two pets and I surrendered, you need a rabbit faster than Lilly.