THIS IS RIDICULOUS AND IF YOU CAN'T FIX IT BY 5PM TODAY I AM CANCELLING MY SERVICE
I'm sorry to hear that, sir. As I've explained we can't do anything about your problem, and I've also explained why. Since we're unable to fix it, I'll cancel your account as you requested. Hang on a moment and I'll get your confirmation number.
It'll only take a moment, sir.
WHAT DO YOU MEAN YOU'RE CANCELLING ME
You said that if we can't fix the problem you're cancelling. We can't fix the problem, so I'm cancelling you.
OH NO NO NO WAIT A SECOND YOU CAN'T DO THAT
I can, and I'm doing it right now.
BUT I DON'T WANT TO CANCEL
You just told me you did. Are you saying you wish to remain a customer?
YES, LOOK, I JUST WANT THIS THING TO WORK
And I've explained to you why the problem cannot be fixed by us, and told you who to contact / what to do / why it's not even an actual problem. So you have a decision to make: You can follow the avenues I've suggested, or you can cancel, or you can stay with us, but we can no longer address this issue for you, and no more calls about it will be accepted. I'll leave it up to you to decide what you want to do. Anything else? (Make it clear you're wrapping up the conversation.)
UH OKAY I'LL SEE WHAT I CAN DO, THANKS, BYE
And that's it. Letting customers push you around is a sucker's game and calling their bluff is a highly effective tactic to get them to stop wasting your time. In my experience, 90% or better of these customers never call back about that problem and never cancel. You just have to make it totally clear that their one "bargaining chip" is useless, at which point they have no further ammo. Remain polite but firm.
Thankfully I don't have to deal with customers anymore, but for a while there, I was the "escalations guy", meaning any ticket that went through the normal tier two guys and wasn't fixed would get bumped to me, and only me, where my job was to put an end to the situation once and for all, either by fixing it or by telling the customer why it's not going to get fixed. I found this strategy to be the best.
A verbal contract isn't worth the paper it's written on. -- Samuel Goldwyn