Protecting Tires from House Cats
Have you thought about a bike cover or bicycle wheel covers ?
This seems the most logical form of protection for little outlay, the most effective method would be to move the bike out of reach of the cats but if that’s not possible then a cover of some sort may suffice.

If the cover isn’t suitable or the cats destroy it, you could easily make a bike stand from some 6mm mdf, make it wide enough to stand the bike in but tall enough to cover the wheels
If I was on a PC i would draw up a quick image / diagram of a simple stand.
If you have hydraulic brakes do not stand your bike upside down.