Success! Following this guide and the gasket replacement guide saved having to buy a new watch. Like Thomas mentions in his review, the screwdriver was difficult to use. I eventually got it to turn the tiny screw by holding the watch from the bottom rather than just pressing down with the driver from the top.
A question: should I reset my watch after replacing the battery? Or will whatever throttling WatchOS put in place reverse itself?
I bought iFixit’s battery replacement kit and the gasket which I received had adhesive on both sides, just like in this guide.
The repair kit does comes with extra gasket adhesive, which confused me at first. The extra adhesive is included in case you manage to open the watch without damaging the original gasket.