What are the constraints for an IV using AES in CBC mode?
The IV for a block cipher in CBC mode must not only be "uniquely used for each message encrypted with the same key". It is usually assumed to be indistinguishable from random by an adversary.

If the IV is predictable, some attacks apply. For example, if an attacker is able to choose plaintext messages with prior knowledge of what the IV will be for this chosen plaintext, she can determine if a plaintext block in a previous message matches a guess; that's simply by submitting a message which first plaintext block is the exclusive-OR of that guess, the IV that will be used, and the ciphertext block preceding the block guessed (or the corresponding IV if the guessed block is the first block); then checking if the first ciphertext block for the chosen plaintext matches the ciphertext block for the guessed plaintext (there is a match if and only if the guess is right).

If I understand it correctly, the scheme proposed avoids transmission of the IV, replacing that IV with a (never transmitted) value obtained by enciphering a unique but predictable value, with AES and the same key as used as in the rest of the message.

This is not quite as good as a random IV. For example, an adversary temporarily gaining the capability to encipher blocks of her choice with the key (or equivalently, temporarily gaining the capability to dynamically choose plaintext blocks as a function of the previous ciphertext block within the same enciphered message) can use that capability to compute IVs that will be used in the future, in turn making it possible to verify guesses on messages encrypted after she had been denied that arbitrary block encipher capability, if she still has the capability to encipher chosen messages (having in-advance knowledge of the IVs, she proceeds just as outlined above in the regular CBC case). By contrast, with a truly random IV, the temporary possibility to encipher arbitrary blocks does not compromise the security of later messages.

Admittedly, the threat model I have been considering is quite hairy, and does not match the one usually assumed in the standard proof that CBC is IND-CPA (where the adversary has the capability to encipher arbitrary messages, but never to encipher arbitrary blocks). I fail to tell if the proposed scheme is IND-CPA, or not (that's an interesting question; an interesting variant would use decryption, rather than encryption, to prepare the IV).

Note 1: It is easy to make an implementation goof when trying to avoid reuse of a predictable value, especially if the key survives power loss.

Note 2: The scheme as described in the question does not insure integrity. That's a major practical issue.

Note 3: I fail to understand how the proposed scheme can be considered inefficient, and how "hash" crept into that consideration.