Dumb question but have you even called your insurance company and asked if you'd be covered without rates going up? We're covered for hitting animals here. I hit a moose, didn't have collision. Was gonna pay to fix it myself but called my insurance company told them what happened and asked if I was covered and if my rates would go up. They fixed it and assured me my rates would stay the same even though I didn't have accident forgiveness coverage. Unless I start hitting animals for sport rates wouldn't go up. That was years ago and my rates only go down.
And out here, if you have more than $2000-2500 worth of damage you're supposed to get a police report. Supposedly some bodyshops won't even touch a damaged vehicle unless you have a report.