You need to speak to a lawyer, not just your manager. Don't sign or agree to anything until you speak to a lawyer.
You should consider emotional and punitive damages. Emotional covers the fear you now have over doing your job. If it's been an intrusive thought since then it's arguably minor PTSD (especially as you were bitten twice). This is not a sign of weakness, it's a normal stress response to being attacked in the course of doing your job. Any therapy you need or later treatment should be covered by that dog's owners. Punitive is the court's punishment for these people allowing this to happen to you and your co-worker, to remind them they've done something wrong and to ensure it doesn't happen again.
You should consider emotional and punitive damages. Emotional covers the fear you now have over doing your job. If it's been an intrusive thought since then it's arguably minor PTSD (especially as you were bitten twice). This is not a sign of weakness, it's a normal stress response to being attacked in the course of doing your job. Any therapy you need or later treatment should be covered by that dog's owners. Punitive is the court's punishment for these people allowing this to happen to you and your co-worker, to remind them they've done something wrong and to ensure it doesn't happen again.