Haven't seen the video, don't want to. But in defense of the driver, sounds like a couple dogs running around as she's driving off, and although I haven't seen the video, maybe there was a blind spot where the dog in front kept playing a dangerous game of chicken with something much larger than itself. Gravel road, bumps are to be expected, so...
It sucks, and I know if I hit an animal, I'd feel terrible. I've hit birds and felt bad. But assume positive intent. Either the driver didn't know, or was afraid to stop and have the other aggressive dog do something.
The owner of the free-range dogs will hopefully acknowledge that this is a very likely side effect of having dogs that not only roam free, but also like to chase (and get in front of) moving vehicles. If they don't want their pets being run over, there are some changes they can make too, so don't pin this on the driver. Even if the driver had stopped, the outcome would have likely been the same. Arguing about "who pays for it" may end up sounding rather callous to an outsider, or in hindsight.
Fedex or the driver could turn around and sue the dog's owner for "mental distress and anguish" that was caused by them hitting a dog, or something like that. It goes both ways, and the best solution would be for everyone to realize that, under the circumstances, this was bound to happen sooner or later, and learn from it.
Fortunately as you've seen from this message board alone, you'll have some kind of idea what the general public may think of this, with people coming down on the dog owner's side, but also on the side of the driver. I'd say you were wise to take the video down just because it is a polarizing type of event.
My opinion: share the blame, call it a done deal, and move on.