I would design the elevator car to be self-contained, rather than pressurizing the entire shaft.
Actually, it would be more like riding in a pressurized moon rover. Then, using wheels or rollers on opposing sides of a track to act as guides, some kind of hoist, or maybe just a linear inductive motor using the track as a stator plate.
The advantage to the linear induction motor is simplicity- if you’re building the track/guides out of metal any way, then, it’s just a few more components and you have a way to go up, go down, and brake.
In any case there’s really no reason to have a pressurized shaft.
The value of the cave is it’s already there, with no need to excavate that incredibly annoying moon dust. Which, to be clear, is a horribly hazardous material.
Because there’s little to no natural erosion, it’s sharp and splintery and highly barbed. It’s also magnetic, so it sticks to anything made of iron or steel; and for example probably causes something akin to silicosis.
It’s basically the reason that NASA and others are looking at pressurized rovers with suit ports for hopping out instead of an airlock and space suit design. It significantly resizes the amount of moon dust coming in.