Have you tried brushing your cats twice a week? This will cut down on the amount of shedding and the cats usually enjoy this.
And they are, after all, in charge.
“Photography is a complex and fluid medium, and its many factors are not applied in simple sequence. Rather, the process may be likened to the art of the juggler in keeping many balls in the air at one time!”
Ansel Adams, from the introduction to The Negative - The New Ansel Adams Photography Series / Book 2
One of the tricks is to close the door to the shower and turn on the shower HOT for a minute or so. Then shut it off, hang your film in the shower stall with curtain closed and door closed. If you can, close the AC vent also. It takes a while to dry but moisture in the air takes down the dust and possibly cat hair also.
I have 4 dogs (two are quite fluffy - shepherd mixes) and this works. It also gives me fairly flat/straight negs (less curl). Luckily, we have three full bathrooms and one stall shower goes completely unused except for film drying.
I built a film-drying cabinet, which solves the problem entirely. Think of a wooden box about 1.2m high (room for 120 film but not 220 or 135 without cutting rolls) and just big enough to fit a fan in the top. Car-engine air-filter below the fan, films hung up below that and accessible via a front door, vent holes in the bottom, door sealed with rubber strips. Air blows slowly downwards through the filter, past the film and out. Not a speck of dust or hair on the film ever since I built that.