Need advice - Herpes

About a year ago I was diagnosed with HSV 1 genitally (herpes) at a clinic. I asked the np there if I should disclose to every sexual partner and she said no that I probably won’t get another outbreak and that it’s different than HSV2.

I got pregnant on a one night stand. I was planning On telling my OBGYN I started going to right away but the father of the baby is really involved and was with me at the appointment when they went over my medical history. So I lied (I know it was wrong). Now that he can’t come to appointments bc of the virus I want to tell my doctor to protect the baby because I read they start antivirals around 36 weeks. I’m 31. Im still worried about how they would keep this confidential if he was in the hospital with me the whole time I give birth and after. They would obviously check me or talk about it. Torn. I don’t wanna put my child’s life at risk because of embarrassment.