The warning bells for me are that he was only interested in your child when it came to influencing you. Then has blocked you from contacting him across all platforms. I would be wary that once your baby is born, that he may try to use this to 'be friends' again.
Personally, I would start thinking two steps ahead. If he's pulling these sorts of childish stunts at this point you need to anticipate it may not change in future and take steps to protect yourself. I dont wish to upset you, but I have known of situations where the father didnt want the baby, ignored the mother, then once the baby was born, used it to make the mother's life hell by taking her to court, refusing to pay, DNA etc. If he is being a dick now, you need to have eyes in the back of your head.
Labour is completely up to you- youre the one giving birth and you need to feel comfortable and completely supported whilst giving birth. Do what is right for you, you dont need to be worrying about someone else's feelings during that time.
Letter sounds like a fab idea. I would write it, photocopy it, and send it via registered post- keep the proof of postage. That way if he does take you to court, or you take him to court, you have proof that you have made the effort to establish contact between him and your child and you are not being obstructive. Keep it cool, calm and clinical.
Get the CMS. You both made the baby and kids are expensive, trust me!! 