I agree with the others. I think you have enough "commitment" going to a wedding shower, baby shower, and visiting the baby.
From another "bride" POV:
I was just married 10 months ago. I alerted my family with save the dates LONG before my wedding - so most of them were there. But if I hadn't, well, I got married in August and expect that many people would have been on vacation! Lots of people didn't come to my shower due to their vacations, but that's ok, because it was in June and they only had a month's notice.
They are the ones doing the quick wedding, so they KNOW that people won't all be able to come - as others have said, if it REALLY had mattered that you were there, you would have been asked if the date was OK.
They will understand. Especially if you still attend all the other things, and if you send a gift. Other family may talk, but they will understand. No one will actually care on the day of the wedding... just one less person to talk to, that's all.
It's not like you're ignoring your family and their blessed event. You will still be there for the OTHER blessed events around this one, and you are still acknowledging this one.