You might want to consider using a french cleat to hang the mirror. It would be fairly solid but allow the removal of the frame and no screws or plugs would be visible. The screws would be in the hidden mounted part of the cleat.
A cleat at the top is all you will need. The weight of the mirror will generally be more than enough to keep the cleat engaged. To remove the mirror you need pressure straight up, pushing in will not disengage the cleat. You can make the cleat out of just about any material, wood, mdf, or even plastic. For small units I have made them out of 1/8" plastic.
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