I've gotta join the "cold-hearted b*itch" club on this one, unfortunately. You have to choose between the two.
Some of us aren't going to lie to you, and we don't want to give you false hopes.
Reality (which SUCKS) is that you're giong to have to pic between the two. Even if you got excused from that day somehow, it's probably going to hurt you (regardless of whether or not it should). Also, would it be possible for you to even miss just one day and be back for the next? She's in Houston, you're in Baton Rouge. It's not exactly a 1 hour car drive away.
Are you going to actually be a part of the birth, or will you be in the waiting room with the rest of the family? I know they've gotten more liberal about who's in the room during the birth. But being that it's a c-section, i'd imagine that they're more strict because it's actual surgery? Someone who's done this before could answer that.
Your future neice and nephew is going to be on the rest of this earth for many many years (baring anything extremelly unfortunate). Seeing him or her a few days later is not going to be that much of a difference. They're not going to look or act much different. They're still going to be an adorable bundle of joy who sleeps all day and cries it's head off intermittently
Rush isn't going to be there a week, a year, a decade later.
Bottom line is that LSU is a cuthroat rush. There's a massive amount of girls, and they'll make cuts however they can. A girl missing a party, even if excused, is an easy cut.
Few, if any groups COB. Rushing next year isn't exactly much of an option either. Hate to sound dramatic, but if you want to be in a sorority, you're going to have to do it.