Still waiting on Apple to allow beta 3 builds on TestFlight, but in the meantime I've finished most of the iOS 10 work and am starting to dig into watchOS 3. Found an explanation in the watchOS 3 release notes for why the complications aren't updating:
Periodic updates have been disabled for WatchKit apps that have not adopted the new handleBackgroundTask API available in watchOS 3. Please adopt this new method to ensure your app is given periodic updates.
So in other words, the app has to be updated for watchOS 3 for stuff like complications to work at all.