You can use the RSA key encapsulation to establish a random symmetric key. Then (in the same protocol message) use this key to protect the data payload (i.e. using some AE/AEAD mode you mentioned).
The RSA key encapsulation should be Padding Oracle Attack resistant (as it does not use any padding).
Another advantage is that your payload length is not restricted as in the pure RSA case.
This scheme provides payload confidentiality (an attacker without server private key can not read the payload) and integrity (the payload is received exactly as intended by the client or is dropped by the server as invalid), but does not provide any sender authenticity (anyone knowing the server public key can generate valid messages with arbitrary payloads).
Sender authenticity might be addressed by signing the payloads.
Desclaimer: I am no crypto expert, so please do validate my thoughts.