Is there a way to provide proof of batch RSA security?
RSA public key encryption should always be used together with a randomized padding scheme, such as OAEP or PKCS#1 v1.5. Only if you don't use such padding and encrypt the same message with a sufficiently high number of different public keys, an adversary might simply apply the Chinese Remainder Theorem to the cipher texts, and then calculate the E:th root of that value (E being the public exponent used by the RSA public keys).