For AES CBC, can I encrypt the IV with AES ECB and the same key and include it with the message?
It actually leaks information. You are sending:


Encrypted IV: $AES(k,IV)$
First ciphertext block of CBC: $AES(k, M_1 \oplus IV)$


Eavesdropper can observe whether the two blocks are equal, which happens iff $M_1$ is all zeroes.