How can RSA be used to compactly sign and encrypt? How can RSA be used for digital signature with message recovery and encryption, providing transmission of a small message with confidentiality and authentication simultaneously, small cryptogram size, and little or no reliance on symmetric/hybrid encryption?
Re-using RSA key pairs for both encryption and signing is catastrophic, since it leaks the private key. So two pairs of keys will be needed.
RSAES-OAEP is a system that allows RSA to encrypt short (shorter than the modulus) messages. It's slow, and the length restriction is a big limit, but it does work.
RSASSA-PSS is a system for signing messages using RSA.
So encrypt using RSAES-OAEP and the recipient's encryption public key, then sign the ciphertext using RSASSA-PSS and the sender's private key.
A better solution is to use static-ephemeral ECDH to  create a shared secret between the recipient's static key and the sender's ephemeral key, then encrypt the message using an AEAD like AES-GCM-SIV, and sign the whole ciphertext+IV+Tag using RSASSA-PSS.