I think that if Serena feels she is owed a baby she feels she is owed it just by virtue of being Serena Joy.
As June said (about the econo family but it fits) Serena "If I'd gone to the right church, if I'd played my cards right...if I'd known I was supposed to be playing cards."
Serena, in her own mind, has done everything right, and for that reason alone she is owed a baby. Any baby. She campaigned for Gilead, she has schemed for Gilead, she has aligned herself with terrorists for Gilead, so if anybody is owed a baby in Gilead she will think it's her.
I'm fairly sure that when Nick and June had sex for the first time Serena was in the room with them. It was her idea, she took June to Nick, and although she didn't hold her down as she usually does, she was right in there with them all the same.