Using HMAC as a nonce with AES-CTR encrypt-and-MAC
What you're describing is pretty similar to the SIV block cipher mode. It also uses a deterministic function of the message to derive the nonce for CTR encryption. Under some pretty widely accepted assumptions about HMAC-SHA256 this is a perfectly fine way of achieving deterministic authenticated encryption. It doesn't meet IND-CPA (as you pointed out) but if this is acceptable for your application, you should be fine. 

Disadvantages are mostly in terms of efficiency. The mode you propose requires two passes of the plaintext to generate a ciphertext, but GCM only requires one. A mode like OCB also only requires one pass for authenticated encryption, but is parallelizable as well. The impact of this is, again, dependent on your application.