If I park my car on the neighbor's grass, and they shoot a bb at it and it breaks my window, and further I'm unaware of it, so the rain ruins something more expensive, the neighbors can be legally held responsible financially for the cost to repair my window, and anything that was damaged due to the damage of the window. They had legal options (towing, police, etc) to take care of the property issue, but instead decided to damage the vehicle.
The $4,000 fine was paid to the STATE. NOT the owners of the dog, who doubtless paid hundreds and perhaps thousands of dollars in an attempt to save the dog's life, and then to pay for cremation.
They are out of that money because the actions taken by a neighbor who had other options to remove the dog without destroying property, or taking an action which COULD REASONABLY lead to injury or death - an airgun is still a lethal weapon in the eyes of the law, though it's not as restricted as a powder actuated weapon.
Should the dog have been on the other person property? No.
Should the other person have shot at or even near the dog with a BB gun? No.
Should the person who killed the dog be financially responsible for the costs of their actions? Maybe.
Clearly the court has already shown that the person did NOT have the RIGHT to shoot at the dog with a BB gun, even though it was on their property.
Now the decision is: If he didn't have the right to do this action, is he financially responsible for the consequences of his action?
I'm reasonably certain that the answer is yes - he did something illegal, which cause another person to suffer financial harm, and those costs should be repaid.
The question of emotional distress is more difficult, and I don't know what the lawsuit is asking for in that term, it looks like they might be claiming emotional distress, but only asking for the costs to recompense thier financial loss because there is no other easy to understand method of suing someone in this particular circumstance without claiming emotional distress.
If they are asking for more than what they paid to buy, care for, treat, and lay to rest the animal, then emotional distress certainly does come into play, and IMO I think both sides have suffered enough to worry about that.
Personally, though, I wouldn't pursue the lawsuit at all. Way too much acrimony involved, the emotional turmoil of a lawsuit would be far beyond what she's suffering now.