I appreciate all of the various opinions.
@Robert Nason intuits that "It doesn't seem to bother you at all that your tenant has loud parties and a constantly barking dog." I did not include the complete history of the matter but as I explain in the chain above I have reached out to the tenant to seek her cooperation in communicating with the neighbor about his allegations. The issue I am exploring is not my feelings about alleged bad behavior but my legitimate responsibility. As @John D. notes, there are competing rights at hand.
Another wrinkle is that I reside faaaar abroad so I am not in a position to be an onsite arbiter of this matter even if certain members of the forum would conclude that I am responsible to solve the problems between my tenant and her neighbors despite the existence of specific City authorities who handle these problems professionally, safely and impartially, for a living.
@Christina Yoon the neighbor raised the matter to my attention a couple of months back. I communicated urgently to my tenant and she promised to handle the issue personally. My neighbor then went quiet for two months then out of nowhere has now threatened to sue me (probably a toothless threat but I do take it and the underlying issue seriously). So the neighbor had provided insufficient notice of the ongoing nature of the issue to me -- plus has failed to take reasonable self-help measures. My neighbor has specifically mentioned that he does not want to call the Police in regard to the parties because he doesn't want to be "uncool." Um... okay.
As to the lease... the tenant listed an essentially barkless Whippet on the lease and paid a deposit. The barking dog, she asserts, is a visiting animal belonging to her boyfriend. Her position is that it does not bark much and thus cannot be said to represent a nuisance. So she disputes that this animal needs to be registered on the lease any more than a person who visits for a short stint. My neighbor obviously disagrees and disputes all of those facts.
If I seek to evict my tenant she would resist. If I could actually manage to evict her the my reward
-- apart from putting a big smile on my neighbor's face -- would be losing at least 40K in rent (which I could not replace as the house will likely be sold at or near the end of the lease term). The tenant might also entertain legal action of her own for moving expenses and other costs if I push her in a direction not to her liking. She is wealthy and has a lawyer write letters to me whenever she wants something rectified.
As to the questions about how my neighbor knows about the lease terms... We were once friendly. So I disclosed to him that she had a Whippet registered when this first came up. He deduced that the other dog was not supposed to reside there. But this is surpassingly irrelevant as I have no duty to anyone but myself to enforce the terms of my lease and I am free to amend those terms to make them more permissive at any moment.
Under these facts I don't think the case is susceptible of a neat solution.