Actually, the baby doesn't need to drink from a bottle. My younger child never drank from a bottle. He went from syringe/spoon feeds for milk, directly to cups. He used Pigeon's magmag and subsequently, Munchkin's. I must admit I dont't like Avent's training cup. I tried that for my elder child and she had a hard time sucking out the liquid.
Why dont't you start your baby on the straw cup directly? Probably she'll take better to that?