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Support 4096 bit RSA keys for Yubikey 4 #58
It looks to me like the current(?) spec from 2015 doesn't support anything but 2048-bit RSA keys: https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-78-4.pdf
Yubico offers YubiHSM2 device, which would an overkill for what you seem to need (mostly because it costs 10 times more than a YubiKey), but if money is not a concern - it supports 4096-bit RSA, and many other nice things (e.g., ECDSA, EdDSA, PSS, OAEP).
I've been working with it, and am quite happy - but my office paid for it.