they may have a strong case in court against you if they find out the truth. An incident very similar to this happened to my grandmother when I was a kid. My aunt was at her house, with her 2 year old beast pitbull. Her upstairs neighbor's dog walked down the stairs towards the backyard and nudged my grandmothers door open. My aunts pit got up the moment she smelled the other dog and took off after it. She grabbed its head and crushed its skull.(this dog was maybe 20 pounds).
The owners of the dead dog sued my grandmother and tried to have my aunts pit put down. They almost lost if not for the fact that the neighbors dog was technically INSIDE of my grandmothers house and thats pretty much fair grounds for dogs 2 murder another dog.
Which is kind of similar to your case except for the fact that you are lying to cover what really happened. No matter how it gets spun or what the truth really is, you are automatically looking guilty having tried to cover it up and also by not going to the neighbors instantly and telling them what happened.
Well I'm glad your grandmother didn't get sued, but I never even came close to lying to anyone about what happened. I just told them I have a dead dog in a bag and you can pick it up at location X. I was never asked what happened, and I don't think the dispatcher much cared.