Well, in case anyone cares about this for the future, I found out more about the battery in an S6 Active.
I called a phone repair place that seemed reputable online. I asked about my situation. The guy said that with the Active, the only way to get at the battery is to remove the screen, and that requires breaking and replacing the screen, which would cost about $200-$300. He suggested that I'd get more value for my money by replacing the phone.
So sadly I have to say good-bye to my S6 Active. It really was still in tip-top shape otherwise. I probably could've tried to milk it for longer by replacing the battery anyway, but the problem with these phones is that they aren't designed to last very long. We had just replaced my wife's S6 a couple months back because the GPS was going out. Fortunately, that makes it suitable for a back-up phone if anything should happen to our current phones. I was able to turn in my dead Active for $30—doubled to $60 due to a promotion on the S9. So there's a little something.