We live in the country, there are several properties that have free ranging chickens. Last week, there were pigs in the road. We are the only property who has even put a fence up towards the front. To finish fencing our acres would cost several thousand more and we just couldn't afford that this year.
These are turning into extremely expensive eggs!
The large single piece bumper does jack up the repair price. That's unfortunate. You should still ask for two estimates if you plan on paying for damages. Had they hit a wild animal - raccoon, deer - their insurance would pay for it less their deductible. But, since they know who owned the chicken, they obviously want to collect from the owner so as to not use their own coverage or be out the deductible.
There's really no way to prove they were speeding at the time of the incident. Even if they were speeding, domestic chickens should not have been in the road. Thankfully they didn't swerve and go into a tree or another car or you could be looking at Bodily Injury damages as well as Property Damage.
You need to keep your chickens from the road to prevent another similar situation. If you can't fence the property at this time, I'd suggest making them a paddock using plastic chicken netting. It comes in rolls, is fairly inexpensive, and can quickly and easily be installed. You use metal fencing rods to secure it. We use zip ties to make sure it's attached securely to the fencing rods. It's not much for keeping predators out but is great for keeping chickens in, which is what you're looking for.
Were it me, I'd get the two estimates and then decide if I could afford to pay it from pocket or if I needed to turn it over to my homeowners liability coverage. I don't know your insurance companies underwriting rules on backyard chickens but I do know that some companies don't like them. Be aware that it could impact your homeowners coverage should your company find that you have backyard chickens and are selling eggs.