Training race horses, we had those kinds of problems with what then was called “the crud”.
We had wood backed brushes and many of them and washed them every day in a bleach solution and put them out to dry on the rail, in the winter in the bunkhouse.
The problem with those kinds of fungus and bacteria is that it is transmitted from horse to horse also in saddle pads, cinches, exercise personnel sides of the boots, most anything you touch when you handle and ride one horse and then another and another.
You can be so very clean and still one horse brings something in and before you know it, there are several with it you have to isolate and disinfect everything.
Since you only have your two horses, it would be much easier to have individual supplies for each one and be very careful to keep them separated and everything disinfected best you can.
Individual sets of cheap brushes and all else, that can take much washing, should work best for you.
Those wet cloths they sell would also help wipe boots, hands and such down when you go from horse to horse.