Agree you need to check what your lease says about repairs, and you need to see if the state law requires you to fix the A/C, and what a reasonable time frame to do repairs is. If it's a habitability issue, you still usually have at least a few days, if not 10 days, from what I've seen of different state habitability laws.
And if it's not a requirement under state habitability laws, normally the landlord can have even a month to take care of something.
Then, you might want to print out that law and provide it to the tenant when you talk to them about following your procedures.
Then, depending on the above, your next step is to put in writing exactly what the tenant is supposed to do if they need a repair done.
Then, you need to write to the A/C company and tell them that you communicated on such and such a date to whomever it was, that they were absolutely not to do any work without your personal authorization, and since they did not honor that, you are not paying that bill. And if they decide to try and collect from you, you will be suing them and asking for punitive damages and your court costs, too.
Be sure and make it clear to the tenant that you will not pay for any bills they generate on their own. That they absolutely must call you to take care of things, or the bill is theirs.
Then, you may want to offer them the "happy clause." Which is, you tell them they don't seem to be happy in the unit, so if they want out of the lease, you're happy to let them out without penalty, as long as they leave the place clean and undamaged.
And follow that offer up in writing.
Sometimes a tenant behaves this way when what they really want is to be let out of a lease. But, they won't come straight out and tell you that. Or they want a rent reduction. If you just offer to let them out of the lease without penalty, it often takes the wind out of their angry sails. They will check for available rentals and they may move. But, in my experience, they usually stay and don't complain as much, believe it or not, once they know they can move whenever they want to. Even after you hold firm on a rule. I've had to do this several times. "Nope, I can't let you do that, but if you're really unhappy here, I'll let you move out."
But, with this tenant, if they know they're supposed to call you first, and they aren't, you need to stop this bullying behavior. So, when they start going on about a sick kid, you can say, "Well, it appears that following my lease is a problem for you, but I can't afford to let you call people directly and just send me the bills. So, if you are not happy with the unit or the repair procedures you agreed to in the lease, maybe we ought to just terminate the lease. You can move out without penalty, and you'll get your full deposit back as long as you leave the place clean and undamaged. I can even give it to you on your move-out day, if it passes inspection. I'll be happy to put that in writing. Why don't you think about it and let me know by next week."
I wouldn't let them suck me into anything about the sick kid. The issue is how they handle needed repairs. Regardless of the reason. And if following your rules is not something they are willing to do they can pay the bills themselves, or move out.
And as far as how you approach them, you can be amiable, but with a firm message. "I know you're worried about your sick kid, and I'm really sorry your kid is sick. But, I am not made of money. It would be great if I could afford to let you call people to do whatever you want and just send me the bill. But, I can't. So, I'm afraid that if you call people to do work directly from now on you will have to pay those bills. And if that's not going to work for you, then I will work with you so you can move out as soon as you find another place."
The message is that you are not going to let them bully you. And giving them an "out" should take most if not all of the wind out of their bullying.
At least, this has worked for me.
