first thing is wet the hay, today, it will help if its an allergy, and if the horse is older to soften the hay, and monitor cough.
if you want to stay stay just take a load of no notice of ym, she might leave, just say yeah yeah yeah to keep her happy, don`t react and wait a while to see what happens, if it gets worse you can`t beat telling it straight to them, say the authentic truth of how you feel and how they are personally spoiling your ownership of horses with their unprofessional and spiteful attitude.
i would ring the vet concerned, and discuss the situation with her, and point blank ask her why she treated the horse without your consent, as this was not an emergency situation, and tell her what your own vet said, also point out that you feel bullied at the yard, you have a right to express your feelings she might be surprised not realizing how you feel, but you need to offload some the distress that you are experiencing, and also say you now have two vet bills - and two diagnoses so would like know which is correct, are you to call in a third vet? ask her what she suggests, and ask should you really be giving your horse doses of antibiotics for a suspected allergy and is she certain it is an infection etc, if so what was the horses temperature? all this you would have know if you could present.
if the price goes up and you want to stay, looks like you have to pay.