wow that is wild well coconut oil is a natural antifungal and antibacterial so i would definitely be supplementing her with coconut oil and it does wonders for the coat - my dd was on it for a year and no bacterial issues along with the probiotics and since yours was prone once you get this all cleared up with vet I would recommend those two things
Ketaconazole is an antifungal drug see below
Ketoconazole: Antifungal Medication For Cats & Dogs - 1800PetMeds
Be very careful if they prescribe this drug and make sure they do not mix it EVER WITH IVERMECTIN. I just want to share this with you as a dermatologist had my friends yorkie on those two drugs and her dog went into a coma so NEVER mix ivermectin and ketoconozole and not alot of vets know this so i am sharing it as I learned this and what happens is with ketaconazole is it breaks the blood barrier wall allowing other drugs to shoot to the brain so you have to be very careful if this is the drug they give as this is the only drug i know of for yeast infections. The above is how it kills the mites as they do not have a strong blood barrier wall like dogs. Do you give your dog Heartguard for heart work if so that has ivermectin in it - not a large dose like off label ivermectin but it is still in the drug so discuss it with them.
The reason your dog got worse on pred is because pred breaks down the immune system allowing infections like bacteria and fungal to get worse as the body does not fight it off like it normally would without it. This is why dd has spun out of control as i have for 2 years supplemented with coconut oil and probiotics and no issues then i stopped and she has been a mess with diarhea and staph infection so i know it works for sure now.
This is a sign of a compromised immune system as she could not fight this thing on her own prior to the pred and then suppressing the immune system made it multiply - yeast and bacteria are nasty and they grow in sugar and everything you eat turns to sugar so it just keeps manifesting itself making things worse - dogs with a healthy immune system are equipped to fight this stuff off though but dogs who do not cannot just like a dog with a healthy immune system can fight off inbalance of mites in skin and dogs with a compromised immune system cannot and why demodex is such a tough thing to get rid off for dogs with a poor immune system.
I hope you are enjoying your glass of wine and i hope it is WHITE lollll have a good night