Don’t tell them to go to the Customer Support.
Tell them to go straight to the Police and to a Lawyer so the CS knows authorities are involved and they are to take serious action.
The police will maybe even contact the CS if they deem it necessary.
Hollow threats at someone’s safety, life and family are a serious crime that should be punished. Not to mention that terrible, terrible things in history happened because no one acted on “hollow threats”.
The first thing to do is giving power to the perpetrator by covering in fear and letting him/her control the victim through blackmail, extortion and abuse.
Police > Lawyer > Customer Support > Social Media > The Press.
Contact them ALL, get them ALL involved, track the psycho down and make sure he/she never, ever attempts such a gruesome act in the future.
I must also say that as someone who they came for help in a situation as terrible as this, your position is very ungrateful, considering the implications and consequences your advice may take.
However, even though its the Clients duty to make the decision for themselves - I strongly suggest you to advise them to do the above instead of giving in to the extorters.
If they sense weakness and fear - as someone highly familiar with criminal psychology, I can assure you 100% that this won’t be the last extortion they attempt. It’ll only get worse.
The only valid response is to furiously confront it with all legal possible means.
Authorities, Customer Support, The Press, The Public - get all involved and demand IP tracking, sanctions from Fiverr and legal sanctions from authorities as well.
My two cents.