Hi I'm sorry you are having to deal with all of this. It sounds very stressful.
Firstly, are you aware that normally you and your ex would be expected to attend a Mediation session (individual MIAMs (Mediation Information and Assessment Meeting) and then, if deemed appropriate, together) and it would only progress to court if mediation were unsuccessful or deemed inappropriate?
There are instances where mediation is not appropriate (for instance, if there has been abuse) but usually a judge would expect mediation to have taken place, or for a mediator to have signed a form confirming mediation was not possible.
I think have a look at La Leche League for advice and potential information about breastfeeding and contact cases. Unfortunately you will see that not all judges and Cafcass are up to speed on this stuff so arm yourself with knowledge www.laleche.org.uk/breastfeeding-contact-cases/
I am sorry to mention it here and don't feel any need to respond on this point but if you have been a victim of abuse where your ex is concerned, or think you might have been, then there is a chance that, depending on your financial situation and whether you would have any "evidence" (from eg a GP or support worker) you would be eligible for Legal Aid.
In general, I really recommend Rights of Women too: volunteer solicitors who offer telephone support for free for those in need. It's very hard to get through on the phone but worth persevering. Women's Aid could also offer you support, I'd have thought.
Very best of luck and I am sorry that these early months are being marred by the stress of this.