So, like you said, you told him he can have 45mins a week, and that has to be supervised by you? I'm not surprised he's wanting to go to court to sort out what's reasonable tbh.
Your representative is paid for by you, they will of course agree with you. If you go to mediation, which you will have to prior to any court action, you will also have to pay, if you refuse mediation, you will look unreasonable in court. Bear in mind that mediation will not involve you saying that 45mins a week is reasonable, that's my final offer.
From what I can work out you've been apart from your partner about 8weeks, so it will take some time for him to work through the legal maze. If he has his own room in this shared acc then there's no reason he can't have her over night. She should be in your room until 6m anyway. And you say she is bottlefed so therefore there is no obstacle other than experience, which you won't allow him to get.
My neighbour has his 4m old daughter three days a week. Full days too, he's never lived with the childs mum, so I would imagine your ex would be able to have her a lot longer. How would you feel if he had your child and only allowed you 45mins a week, with no time alone with her?