It seems to me that you and your mother have attempted doing what I was going to initially suggest which was to talk with the neighbor to arrive at a resolution. I'm sorry to hear that it did not work out well.
As an officer, we respond to a variety of calls involving neighbor disputes or disturbances: complaints of neighbors pushing or shoveling snow in front of mailboxes, in front of neighbor's homes, onto the sidewalk, or into the street*, and complaints of unrestrained pets, or complaints of pets defecating in neighbor's yards, and noise complaints, parking complaints, complaints of unsupervised children in neighborhoods or unsupervised children playing in neighborhood streets.....and the complaint list goes on. Most nuisance complaints such as noise, parking, and animals can be handled with a warning or citation issued by law enforcement, parking control, or animal care and control. Also, neighborhood associations can impose fines or restrict privileges. Reports of unsupervised children or children playing in the streets can be made to police and/or CPS (Child Protective Services). Outside of that though, the issues or complaints can be difficult for law enforcement officers to respond to because there isn't always a criminal offense or violation occurring. So the goal is to attempt mediation or to arrive at some resolution to restore peace among the neighbors. That can be difficult because as you have seen first hand, some people are simply not willing to cooperate or communicate.
My condolences on the passing of your father. It's unfortunate that this dog, which has brought your mother comfort after the passing of your father, has caused conflict for her with the neighbor. Perhaps if the neighbor knew that this dog is more than just a pet to your mother, they might be more understanding and tolerant.
My best advice at this point would be for you or your mother to keep your dog away from the neighbor or your neighbor's yard or home. Also, be mindful of animal control provisions, rules and regulations, specifically the one identifying an animal as a public nuisance which has to do with an animal barking, whining, howling or making other sounds in excess or continuously for 15 minutes.
*Just one last piece of information for any other readers that might be wondering about shoveling or pushing snow onto sidewalks or city streets, that is a violation of City Ordinance 72.81 - DEPOSIT OF SNOW AND ICE FROM PRIVATE PROPERTY ONTO SIDEWALK OR STREET PROHIBITED. It reads:
It shall be unlawful for any person to deposit, put or place or to suffer, allow, or cause to be deposited, put or placed any snow or ice from any private property upon any public sidewalk, street, alley, or within five feet of any fire hydrant.