If it shoots BB then it's not really up to the task, your most affordable options for the US wuld probably be a Daisy Powerline 880 or Crossman, they'll pack the power needed for a clean kill and can be found in places like Walmart.
I think Pyramid Air has a forum, that might be a better bet for getting the answer as it's US rather than UK as mentioned we can't shoot Starlings here and Grackles are not native.