The app itself is objective-c, but all the animations were pre-rendered from Hype documents. In fact, it was one of the reasons we added PNG Sequence Export to Hype .
Pastime Baseball was a watchOS v1 app, which basically meant it is only capable of having a small amount of UI that needs to be in constant communication with the iPhone. The original challenge was figuring out if a game could be run at all, and then there were many challenges to get reasonable timing.
watchOS v2 and later allows for actual code running on the devices. As you've found out though, it does not support web views. You'll likely still need to render animations as PNGs.
@stephen wrote a tutorial here, but since it is referencing watchOS v1 I expect some of it to be defunct: