whatamess!!
FIRST OF ALL, PLEASE TRY NOT TO STRESS ANY MORE! This DOES stink, but the place didn't burn to the ground and no one murdered anyone in your rental property. These are good things. I had a water heater pee all over my property's living area. Then after repairs were completed, despite my best efforts, someone reported smelling mold such that I had to move the tenant out while I dealt with that. THAT was a health issue.
I'm not sympathetic that the tenant is saying she's feeling unwell when you offered window units. Either she sucks it up OR she lets you put her up in a hotel. But declining your offer of window units only to then complain about the heat is a little silly to me. The other alternative, which would be to deduct some of her rent for the month, as someone said, is a great idea.
As for the jacklegs who installed the AC, I agree that I would have dragged them back over there and had my husband present to deal with any snide comments and ensure they took their work more seriously. It's a shame you have to think like that, but it sounds like that's the reality you're dealing with. I'm not sure how much your contacting your regular AC guy might have complicated things if you do ultimately end up in front of a judge (I'm a criminal attorney, so forgive me for not knowing much about civil). But, if I'm not mistaken, all he did was add some freon (right?), which means he didn't do any work on the mechanics of the equipment. If that's the case, I would hope the jacklegs shouldn't be able to claim that the unit was somehow altered before they were given the opportunity to repair it. (I hope that makes sense.)
Either way, from the standpoint of getting it working, in light of the fact that you are dealing with an inept or unscrupulous contractor (aka jackleg), I would try to convince the tenants to move out for a day or two while you deal with the jacklegs. The back and forth is legitimately stressful for the tenants, which is the reason you (as a conscientious landlord) are stressed in turn. Getting them out of the house will buy you time and the ability to focus on having candid conversations and dealings with the people who came out and took half the night to install a simple AC unit.
Meanwhile, I would also consider researching whichever agency has authority over these contractors and have their complaint number, as well as the number for the Better Business Bureau cued up on your phone if you need to go that route. I know these contractors might not be registered and licensed. But they are under SOMEONE'S authority and should be sobered by the fact that you might call if the problem isn't fixed by a certain time on a certain day.
My recommendation: 1) get the tenant out of the house for 2 days (max) (give them a few gift cards to places like UberEats, Chick-Fil-A, Ruby Tuesdays, PeiWei, or Outback Steakhouse (if you want them to love you forever and forget about the nausea, I recommend you go big)....throw in a gift certificate to Cold Stone Creamery (they're hot, right?) and they will be forever grateful their AC broke; 2) call the jacklegs and have a stern conversation with them, explaining that you would like (expect) them to meet you at the property asap. Be pleasant with them until they give you an attitude, then make a note of their off-color comments and tell them what you plan to do next (in terms of reporting and/or suing).
my apologies for the long response; I hope some of those thoughts were helpful. I am praying for your peace, first and foremost. I know what it is to have your stomach in knots, but you've got this. Sic your husband on them. If he's as ticked as he should be that your stomach is in knots, he should be able to deal with the foolishness.