I make a lot of saw dust downstairs so dust control is a concern of mine. I ended up making my own whole room air filter out of an A/C air handler that we had replaced a year ago. The filter media was obtained from American Fabric Filter - Custom filters, Dust Bags and filtertration equipment and parts
I would think an axial fan with a filter bag over the exhaust end would control cat hair and dust bunnies just fine. You might get by using a shop vac HEPA bag over a small house fan. That wouldn't cost much to try.