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Best choice will be thick metal medium, such as stainless steel.

Probably use a 177x177 QR code with Q or H error correction engraved (deeply). I would probably compress the data and encode it as Hex so it can be stored in alphanumeric mode on the QR. You might need a series of QRs for larger data.

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I completely hadn't considered the idea that QR codes could be made 3D (in fact, now I'm curious how much denser they can get, if you can both read and write them in 3D), but now that people have said it, it seems so obvious! I really like your follow up about using the error-correcting properties of it. A great feature that would be very unlikely to be realized in amateur 3d data implementation.

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