I was able to get my hands on some new batteries directly from Lifeclock One thanks to Jonathan offering to help in this thread, but because of their age they unfortunately look to be exhibiting the same problem as my original battery. (Very erratic power levels; charges too quickly and immediately loses 1/4 or more of its charge after it reads 100%.) But, I do have some experience now with replacing it.
In case anyone tries it, here are a few pointers. The battery will be glued to the inside of the watch back with some adhesive tape. The easiest way to remove/install the battery is to first pry it from the metal back it is taped to, allowing you more freedom to play around without disconnecting anything else. (There is also a plastic frame inside with other components attached which you can keep connected; it shouldn't interfere with the next steps.)
Note the plug for the battery does not actually slide into its designated slot as I originally thought:
Instead, there are tiny tabs on the plug which correspond with little spaces in this slot. The plug actually pulls up out of this slot to remove or is inserted in a downward motion. As long as the tabs line up correctly, it’s very easy to push it into place.
Once the battery is plugged in, you can reattach it to the inside back of the watch. I used ¾ inch Scotch permanent double-sided tape; first I briefly made sure I could close up the watch back where the battery was situated, then I pressed the battery to the watch back for about 60 seconds for the adhesive to take hold before sealing up the watch for good.
I think a brand new replacement battery is the only way to go. But because of the plug situation, it may be necessary to attach the wires of an original battery to a new battery with the same dimensions in order to fix it.