Does ECIES imply authenticity?
I guess the answer is no, as long as you are using ECIES then this protocol does not work - you cannot trust the public key of Bob, which is required for ECIES. 

You could however use ephemeral-static Diffie-Hellman, using ECDH as cryptographic algorithm. Alice would supply the static part as her public key is trusted, Bob may use any key pair. That means that Bob can trust Alice, but not the other way around. This may however be sufficient for some protocols (TLS normally does not perform client authentication either).

Of course, you may just as well use a static key pair for Alice instead of randomly generating it again and again, requiring to set up trust for each key pair creation.