So let me get this straight. It is your contention that the gym should buy you whatever piece of equipment happens to catch your fancy, no matter what the cost to the gym and no matter how few customers would use it? Because there is no way your complaint makes sense otherwise.
It sounds to me like there is no way for the lat pulldown machine to be safely used by somebody with your limitations. That is unfortunate. But presumably the gym is full of machines that you cannot use. Treadmills, stationary bikes, ellipticals, leg press machines, etc. Why not complain about those? You said the new gym didn’t have a hand-cycle for you to use for cardio. Isn’t it time to sue them for that?