(California) Having baby at the hospital, GF has insurance through her work and I’m uninsured. Hospital says baby has to go on Mom’s insurance?

Hello, I have a quick question.Our baby will be born soon and a lady came by to ask about insurance. My GF has insurance through her work with an OOP max of $4,000 and family OOP max of $8,000. I’ve been uninsured but I do have a form of Medi-Cal (I’ve never had to use it) I’m currently unemployed and looking to start a job shortly after baby is born. Our OBGYN is billing at delivery and I’m assuming with that + hospital bill we’ll hit the OOP max. We’re not married, and I’d rather have the baby on medi-cal instead of her insurance so we save the $4,000. The lady here said as far as she knows the baby has to be on Mom’s insurance. From everything I’ve read, we have 30 days to decide baby’s insurance, and also they look at household income which we’d qualify for. I asked about Medi-Cal presumptive eligibility and she said she needs to talk to her supervisor. I want to avoid paying the extra $4,000 and just save that money for the baby. From everything I’ve read I can do this by adding baby to medi-cal. Can someone please clarify and is there anything here at the hospital I need to do or do I have time after she’s born etc? Thank you!