@TomTomBinks,
I was throwing it out there, re how to build a cat enclosure - I spent years as a landscape designer but I've never done one of those and would have to think about the 'how' of it, the strength of the materials, the need to keep as much light and air in the enclosure as possible; if chain link, if I remember, it can be available with the open spaces smaller than usual, not sure of that, but that is more Jail the Cat like. Even regular chain link is Jail the Cat like. Maybe plexiglass is a better idea, for the cat to see out. Not all plexi, interwoven with some kind of mesh for air.
It might be fun to put together something, but I'm still for letting the cat alone. I think your sister is way off about the cat being miserable outside. Way off.
I might let the sister just stew, guessing she won't let the mother down about other things for long. And ehBeth's strong suggestion for having additional people to be able to help her as a senior is right on, in any case. That's an important point.
I get predators exist, and cats are predators too, but the main problem being the cat is miserable out there is pure projection by the sister - unless it is miserable, but it doesn't sound like it. Anyway, it's the mother's call.