Is the interior of an aircraft providing public transport considered private property, I would say yes.
There is no doubt, private property. A train or terminal is public. Some catches though, for example you are in your front garden mowing the lawn in your bikini and I drive past and snap a photo. That is legal as there is no expectation of privacy even though you are on private property as I can see you from a public road.
So in summary, by the letter of the law if crew ask you to stop filming and you don’t then you have failed to follow a lawful instruction and can be removed from the aircraft. Good luck with that.
It’s all in the how right? If I quietly ask the CM if I can swap if the other passenger agrees and make a point of being polite and respectful it would probably be successful. This tool behaved like a child.