So, the threat that they might get evicted again is not an issue.

I regularly get threats of lawsuit. 

I need some help to pitch this to my uncooperative, inherited tenant. This tenant has a steady job, a single mother, and has a kid in private school. In a nutshell, I want her to leave ASAP, possibly use cash-for-keys...

Assuming you're running a professional operation within the bounds of the law, then asking tenants to pay their rent is completely legitimate.The first time a tenant threatens to sue, it can be alarming...but believe me--at a certain point, you've heard these baseless threats so many times, and you're so knowledgeable about the relevant laws and your own rights as a landlord, that you become emotionally numb to most lawsuit threats (again, assuming that you're operating a professional business that's within the bounds of the law).I don't know about the energy audit, but it sounds like just some more blustering the tenant is doing to try to intimidate  you....if the tenant is responsible for the utilities, and they think their utility bill is too "high", the obvious response is: "you have total control over the amount of utilities you consume.

I'm writing in 'Joe Threats and/or Assigns' so if for any reason I am not able to close, I will find someone who will.

If the policy is large enough, there will be little to any threat in losing the property if sued.

But still, the threat of eviction helped get them out at least.And aside from the cleanout and laminate, I was able to get it rented right away.

Your tenants threats are just a temper tantrum.

Can I ask what prompted the realtor to make the threat?

Can I make this specific demand under threat of eviction/termination of month-to-month lease?