This has to do with another question that someone asked, and I got to
wondering if the police department is protected from being sued in
small claims court.

The suit would be to force them to take action against a barking dog.

Answer:  No

Spend the time suing the owner of the dog.

Clarification:  You can sue in small claims court - but it would be
thrown out.  That's not what small claims court is for - "small claims
" refers the amount of money you would be suing for.  You would not be
suing for money.

It established law that the police do not owe a duty to any one
citizen (there is na exception for 911 callers), so you can't force
them to do anything.

Is a civil suit against your neighbor.

Your town likely has a code about barking dogs.  You can contact the
authorities repeadetdly until they take action.

In our last neighborhood, we had a two neighbors, one had 7 dogs, and
would bark all day and all night.  We complained but nothing was done,
so our Other neighbor called 911, repeatedly, eventually the cops came
out and told the woman if she didn't get the dogs inside, they'd
arrest her for disturbing the peace.  It worked, no more barking....

Take the high road!