I can imagine that TSA might be a bit skeptical of a handwritten document (Swedish emergency passport) that the person checking your documents has never seen before, but that's not all the ID you have. You have a valid green card, which they definitely HAVE seen before, and is US government issued.

SAS will want to see your passport when you check in. They'll have definitely seen a Swedish emergency passport before, so no issue there.

Your green card is definitely acceptable ID for TSA, whether for domestic or international flights. Start with that. In the event they do ask for a passport (they shouldn't, because whether or not you can get into Sweden on arrival is SAS's problem, not TSA's, but they might), show them the emergency passport.

Really, you're making a mountain out of a molehill here. Is it possible you'll have a problem? Sure, you could also get into a car accident on the way to the airport, for that matter. That said, I really really doubt you're going to run into any trouble whatsoever.