Agree with what everyone else said about no overnights until at least 1, and you will only have to then if he has been consistently involved and it is considered in DC's best interest, as in DC won't get too distressed.
Also if you are breast feeding, you cannot be made to pump, by him or court. He will have to initially visit with you there (I'd guess until about 2-3 months, longer if you have any genuine concerns about his ability to care, or about DC's distress level) and then once DC is feeding with longer gaps in between he could start taking DC out for a few hours at a time between feeds, but you would need to be nearby incase DC did need feeding.
Also BF is recommended by WHO up until 2 years, so if you are intending to do more natural term BF then overnights/longer than 8-12 hours wouldn't really be possible until 2. Court won't interfere with breastfeeding as long as you are offering reasonable contact, as it is what is best for DC's health.
They are looking at what is in DC's best interest, not into "parental rights" which don't actually exist. Parental responsibility is a parents duty to the children, only children have legal rights in these situations.
There is no point worrying, as you don't even know what DC will be like yet. When it has just been mum from day 1, they will quite often get more severe separation anxiety early on from what I have seen. If DC does get very distressed being away from you, then STBXH is (if he cares about the DC anyway) unlikely to push the point until it will be an enjoyable experience for them both rather than just him being screamed at endlessly by a baby who is scared and resenting him for taking him away from all he is familiar with.
My DD screams blue murder if I even leave the room, and she is 9 months. It's to the point where my own mum doesn't even want her alone, after her having all these plans to take her swimming and have grandma time so on during the pregnancy.
On the hyperemesis note, stress and dehydration both make it worse (as you probably know) so try to force water down in regular little sips, make sure you are eating as much as you can, and relax. Have a bath, sleep in as much as possible, relax infront of the tv when you can and don't put pressure on yourself to be doing too much, the priority is keeping you and baby healthy. The other DC will cope with a few months of slightly less interesting activities/slightly more TV, and your STBXH needs to be ignored as much as possible until the DC is actually here, other than handovers for the other DC etc.
If I were you I would tell him you will discuss plans once DC is born, as you aren't feeling able to deal with extra stress during the difficult pregnancy.