I can't see there any serious problem either?!
We always have to have plastic gloves in our cars 1st aid bag (I don't know wether these are requested in other countries as well) and I think if you feel uncomfortable you could use these...

I think it's really great that you do remove these poor dead bodies from the road, because most people really don't care about them...
It's a good thing you do and I hope you don't stop...so the little furry ones at least can watch from above that someone has cared about them...