Dogs really handle weightlessness fairly well. I often floated my Lab around the cabin of small aircraft. He usually seemed to think he had chosen a defective seat, as he would find another as soon as he could stop floating.
Although I have no first hand experience with the same maneuver and cats, I have been warned to NEVER ATTEMPT this with a cat! Words and statements along the lines of “lucky to have survived” are to be associated with such attempts. Of course, YMMV.
Don’t try it with a cat because they see your head as the most stable platform in the airplane, and under those conditions they find human flesh the best surface to get a good grip on.
Most cat owners have survived the experience of having a loose cat in a car, and don’t ever want to duplicate it, but a pilot might not get a second chance.
Okay...I’m laughing myself silly just imagining a cat in freefall...
Now you have to do this and record it on video...all you have to do is place a cargo net between the the front and back seat so the critter can’t reach you...it would have to be the world’$ funnie$t video.
Of course you would have to wear one of those shark mesh suits to get the cat out of the plane after the ride.