First off, I assume that it is legal to discharge a firearm in the area. It's not in many areas. Working on that assumption, I'm sure it is necessary to have a depredation permit to kill these animals out of season (if they even have a season). You can't just go shooting anything anytime anywhere. Grackles and cowbirds are protected at all times under the Migratory Bird Treaty Act.
I suggest you arm yourself with the regulations. Contact DNR and find out when and under what circumstances it is legal to shoot each of these animals. Then, if you feel you have some legal footing, you can contact your local wildlife officer. Do it by phone or in person, not in writing, and don't take "no" for an answer.
Bear in mind that your neighbor is likely to figure out who the rat is.