Add the 1x2 nailer to the top inside of the cabinet back.

Secure the nailer with glue to the flimsy back and glue and a couple of nails through the cabinet sides into the ends of the nailer and a couple more nails through the the cabinet top into the edge of the nailer. Paint/stain the nailer as needed.

"The walls are plaster and lathe, so no way to use a molly. "

Actually, you just need a longer molly.

If the cabinet goes to the floor for weight bearing, all the all fastening is for is tipping.

A Molly setting tool works much better in plaster walls.
The small teeth on the molly are not adequate to dig into the plaster and prevent turning during tightening of the molly and expansion.

Since there's no room for error in leveling, spacing and lining up screws with those keyholes, I'd go the Mongo route. Otherwise, if you're off by a hair it's a nuisance.