Should I ask for medical fee when I got serious bite by client's dog? If so, what is the best way?

Today morning, me and my wife tried to take a client's dog for a morning walk after he finished his breakfast. When we open the door, the dog suddenly jumps/pulls as hard as he can, then got off the leash(he pulled his head out of the collar). We ran/followed the dog two blocks till he stopped running, and seems clam. We approached the dog slowly, but when we try to put the leash on, he suddenly bites both me and my wife. The bites are serious and deep. (pictures attached)

Timeline:

2/24/2018 Meet & Greet at my backyard

The dog looks a bit nervous, tend to run away from me and my dog. I successfully petted him. The dog's owner says the dog is kinda shy. I think that's normal, the dog didn't show any aggressive sign.

3/17/2018 The dog's 1st boarding day

When I unleash the dog, I realize the leash was too loose, so I tight it up a bit before putting it away. The dog looks more nervous when his owner left. I let him to the yard to watch how other dogs play, to see if he would like to join them. He sits in a corner and tend to run alway from other dogs. So I let him back in house and petted him for half hour and gave some treats to make him feel easier.

3/18/2018 The dog's 2nd boarding day

The dog seems slept well and finished all of his breakfast. Then the bites happend. I think it's dangerous to keep the dog around my family.

We contacted Rover and the dog's owner. They say sorry.

We washed hands with soap water, then went to Urgent care in our area to clean up the wound. Two of my wounds still bleed after we back to home.

After 5 hours, Rover and the dog owner found another sitter to pick up the dog.

Rover modified the fee for boarding the dog to one day.

Caution: images below contain blood.

My wife's hands got bite when she tried to clam the dog and hold him to let me put leash on: http://i65.tinypic.com/mal5ab.jpg (image description) http://i64.tinypic.com/2db8fn6.jpg (image description)

My hands got bite when I tried to put on the leash: http://i66.tinypic.com/23r3b0m.jpg (image description) http://i68.tinypic.com/dh5w5u.jpg (image description)