Thanks for the update. I had a longer post ready to go but I think you’ve clarified things quite well. I’d contact the agent’s broker and lay out the situation emphasizing how you put your trust in them since you were doing this from afar. The cost of the warranty, although now only a fraction of what you’ll be out to replace the AC, seems like a fair deal since you’re both at fault. It is completely unacceptable for the agent to be ignoring you. If you don’t get traction, I’d look into reporting them to the state RE board, negative Yelp review, and that sort of thing.
Also, to clarify for LSLGuy and others, WI isn’t one of those weird states. My wife is a RE agent and we’ve bought/sold a couple houses here. Our experience is pretty much spot on with Sateryn76’s post, and I have no idea where OldGuy is getting his info. I suppose there are some dinky towns with a single broker where it might be the case, but definitely not representative of the state.
Don’t even bother with your agent at this point - call his/her office direct line, get the receptionist, and ask for the managing broker. Lay this all out, and demand to be made whole. If in fact you have it in writing that you demanded a warranty, and the agent didn’t put it in the Purchase Agreement, then you may have an argument that they owe you not just a warranty, but the costs to you of the A/C fix, since it wold have been limited to a service charge ($100-ish) if the warranty had been in place.
However - the closing documents are not the controlling documents. The controlling documents are the Purchase Agreement you signed (???) at the beginning of the transaction. There’s no world in which an offer would be made via text, without an actual contract being executed by you. There was something you signed at the beginning of all this - pull that document and make sure it included the home warranty provision. If not, you may be SOL, and wiser in the future about signing things without reading them.