Those diseases which medicines do not cure, iron cures; those which
iron cannot cure, fire cures; and those which fire cannot cure, are to
be reckoned wholly incurable.
If you do not want your cats to use medicine, have them rely on surgery (iron). Even up to 150 years ago, things we now would treat with ingested or injected medicines were treated with bleeding which is essentially a surgical technique. I could imagine your cats having a medical philosophy that was half medieval barber-surgeon and half acupuncture. That seems catlike and also would be easier to write.
You could go Hippocrates all the way and also have them use fire. I think the principles of moxibustion combined with acupuncture would be great cat medicine and exciting reading.
Nonplant medicines.. If you want medicine but cannot use plants, there are many, many inorganic options, especially if you go back 75 years or more. Calomel is mercury chloride and was used as a purgative for hundreds of years, up until the middle 20th century. Arsenicals were also used for hundreds of years; the first effective antisyphilitic Salvarsan was an arsenic compound. Arsenic still finds use against cancer and certain parasites. Copper sulfate and potassium iodide are formidable antifungals, the latter still used today against certain fungal infections. Gold compounds are effective anti-inflammatories. Sulphur is good against many skin diseases. Salts of potassium, magnesium, sodium, and calcium all had and have medical uses.
There are hundreds of inorganic medicines based on metals, minerals and salts. If you want to go that route and get into detail then I propose you first pick the disease you want your cat doctor to treat, then use google books to research that disease limiting to texts pre-1940.
ADDENDUM: Animal medicine
These plantophobic cats probably would like medicine made out of prey animals. One they could use is salt pork. Substances in the pork can stimulate the clotting cascade. Salt pork or bacon up the nose has been used to stop nosebleeds for a very long time. I suspect the preservation method is not as important except to have some pork handy when needed.