How do I patch a 3'x3' hole in a ceiling made up of plaster, lath, and metal mesh?
The fact that sections are peeling off indicates that there is a structural problem with the overall "sandwich."

In any repair, unless you can overcome the structural deficiencies with an overriding mechanism, you have to strip back the unsound sections until you get to good structure.  With you description , it is hard to see where that is.

You could knock loose all the compromised sections and then try to patch the open areas, but it sounds like the overall lathe-plaster-chicken wire-plaster amalgam may be questionable. You may do this and shortly thereafter find a new section that is failing.

I think that a drywall overlay may be the simplist, fastest and cheapest solution, by far. Whether you strip down to the chicken wire or to the lathe depends on how much of the ceiling is loose/compromised.  If height is not a problem, dislodge whatever is loose, use shim to insure level and then screw up thin drywall over the whole ceiling. DO NOT USE NAILS! If you hit studs, great, but the lathe should be adequate to hold a thin wallboard lamination.