Wifi wont work at all (or at least be very packet-droppy) in this configuration

Hi, I have worked in numerous shielded environments, built one, and am in the process of building a second.

Wifi works perfectly fine inside a shielded enclosure, if both the AP and the client are inside the shield. It should not work across the shield, if the AP is inside and the client is outside, or vice versa. (If that worked, it wouldn't be a very good shield.)

It is entirely plausible, practical, and not even all that hard, to build precisely the environment described up-thread. "Magnetic" paint is not necessary, it just has to be conductive. Ecofoil® Ultra NT® is my favorite shielding material, it's good as a radiant energy barrier (say, to keep your hot roof from radiating heat down at your attic) and as a radiant signal shield. Which makes sense, when you consider that RF is just RF is just RF. Filtered power passthroughs aren't particularly hard (Start with the Delta 20DBAG5 and add some ferrite beads), and if you really want to be snazzy with your data passthrough, use fiber. There are all sorts of cheap-and-cheerful ethernet switches with SFP slots now.

The door seals are the tricky part. Commercial shielded enclosures go all-out with complicated lever-actuated doors that wouldn't feel out-of-place on a bank vault, but I've found that simply sanding the paint off a commercial steel door and covering the bare steel with copper tape, then engaging it with beryllium-copper spring finger-stock around the doorjamb, is sufficient for about 60-80dB of isolation, which is plenty in many environments.

Inside of the cage it'll be fine. It just won't do great traversing the boundary. As long as there's a WAP/antenna inside the cage everything inside the cage will get a signal.