Feeling hopeful, I went back outside, hooked up the charger, put it in AGM mode, and pressed the charge button.
Looks like I'll be hunting down a battery tomorrow.
I'm debating whether to stick with the Optima Red, which is a starter battery, or the Optima Yellow, which is a less powerful deep cycle battery. Starter batteries are designed to use no more than 5% of their power; a full discharge significantly reduces it's battery life. Deep cycle batteries are designed to be fully depleted and recharged without damage, but they produce less cranking power. The Optima Yellow is also significantly more expensive than the Red, and it has only a 1 year warranty, the Red has 3 years.
Deep cycling a starter battery generally voids the warranty, so I doubt I'll be able to get a warranty replacement.