Would exposing a cryptographic hash function's digest (e.g. SHA-3) of RSA private key data compromise the key? If so, what are the possible (cryptanalysis-) vectors for attacking the key if an attacker had knowledge of this hash?
No, exposing such a hash does not compromise the RSA private key, unless the hash function is sufficiently and severely broken.
Of course, you don't need to hash the private exponent to identify the key. You can simply use the modulus or a hash over the (public) modulus to identify the key. This has the additional advantage that that ID will also match the public key. This is common practice, e.g. in the PKCS#11 standard for secure tokens.
You could use a hash over the private key as a proof of possession, but that has the disadvantage that the requester needs to have the private key and/or hash in advance. This is normally solved requesting an RSA (sign or decrypt) operation instead. I've never seen a hash over the private exponent being used as proof of possession.