re: Mirrorless for cats?
>"the left hand, a jingly toy or snack to get the cat's attention."
First thing I would recommend is that you (temporarily) attach a jingly toy to the camera strap.
Second thing I would recommend is one of the new (fixed lens) compacts with a large sensor and bright lens. e.g. the Sony RX100. Although be aware that these lenses are only bright on the wide angle end. i.e. they get darker as you zoom in. Which brings me (back) to...
Third thing I would recommend (in conjunction with the first thing) is what I said originally--an older entry level DSLR--e.g. Canon 600D or Nikon D3000 body only--and a 50mm f1.8 lens, which together should be in the same price range as a Sony RX100, but give you a little more magnification at f1.8 to better frame the cats. (And clip a jingly toy to the strap.)