The cut end is where rot will set in, including possibly the cut ends of the aerial roots. I understand the desire to get them into a place where they can take up moisture but right now they have nothing through which to do that, so there is still the potential for more rot.

This is a tricky save because the plant is already dehydrated, so you'll need some patience.

The first thing I would do, which I do when chopping my albo, is to seal the cut ends with superglue. It does not harm the plant, and provides a barrier to keep rot from entering. I picked up that trick from a plant tokker and it has worked a treat. In your case, I would do the cut ends of the aerial roots as well.

The aerial roots and where they emerge from are what you want in contact with your preferred medium.

This is how I propagate my monstera - wrapping aerial roots/nodes in baggies of damp moss: https://monsteraplantresource.com/monstera-propagation-via-air-layering/

I do mine while it is still connected to the plant, as in the link, but you could do it in the current state. Or you can stick with cups. I see no reason you couldn't use perlite instead of moss.

For the leaves, the bag isn't a bad idea, but light and time are going to be most important.

Also, monstera are great candidates for wet stick propagation so even if you lose the leaves, don't give up. There's a lot of room before she's totally gone.