Why do they have to be so big? That's a ton of magnetic energy!
Use a recessed hole in a piece of mild steel. You'll get the same holding force in a fraction of the size.
The steel acts as a pole piece, carrying the back-side field around to the front (concentrated around the rim and face of the hole), not just doubling the effect of the field (instead of one pole face, it's two) but increasing it significantly as well (instead of flux density diluted into free space, it's concentrated in the pole pieces). Only a steel plate is needed for the opposite face, or you can use the same geometry with opposite polarity for somewhat more holding.
Needless to say, the steel focuses the field, so it's not going out all over the place, but confined mostly to the front face. This also means the grab distance is shorter, but that should be fine for a latch.
Ideal geometry would be something like, say a 2mm dia 1mm thick magnet, in a 3-4mm I.D. hole (centered -- maybe use a stepped flat-bottomed hole to get it centered more easily), with minimum O.D. 6-8mm say, and minimum thickness behind hole / of catch plate, say 0.8mm. So, it can be as small as the original proposal.
If you can't mill/turn flat-bottomed holes, then perhaps these can be found off the shelf as latches/catches?
As for snapping in place, there are zillions of less aesthetic options. Metal and plastic tabs locking into holes; edge catches/latches; slides; clamps; etc.
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