Fixed - Capacitor #37
After a lot of research, I eventually opened up the sub and removed the circuit board. 3 wires connect the amp to the speaker. I took a picture to be sure I remembered how to put it back together. On inspection, it was pretty obvious that C37 was fried. It is a 6.8uF non-polarized capacitor (non-polarized means you don't have to worry about which end is which). I was grateful that it was not near any transistors or integrated circuits so I would not have to worry about a heat sink. The cap de-soldered very easily, but I needed an exacto knife to cut it free of some goo. No problem.
I found the replacement part on Amazon. It was pinned axially, not radially, meaning the leads were on opposite sides instead of on the same side as the original part was. No problem. Axial caps are made so that the leads can be bent to reach all the way around. Once I had the leads pointing in the same direction I cleaned up the holes in the circuit board a little better and removed the char marks from the original part blowing out. I soldered in the cap, reattached the wires to the speaker, and screwed the board back into the sub. Plugged it in, said a little prayer, and turned it on.
It worked perfectly. The sub starts "on" but turns itself back off when there is no signal for a few minutes. That "greeeeep" noise is even a little shorter. :-)