The FIPS 140-3 approves Key Transport Method using RSA-OAEP in NIST SP 800-56Brev2 .
In order to achieve the NIST SP 800-56Brev2 compliance, several checks must be implemented by the module which especially include assurances for the key pair owner (56Brev2 Section 6.4.1) and assurances for the public key recipient (56Brev2 Section 6.4.2). This might not be easy to achieve for a software module, but if I am right, this is now implemented in OpenSSL.
Therefore we consider for the FIPS mode any public key encryption/decryption interfaces non-approved. If we would like to revisit this in the future and certify public key encryption using RSA-OAEP, the above requirements need to be met.