Made an account to share my solution. This is definitely an issue with the Weather app/widget/complications. Here's what I did:
(Please note to have both devices charged to 100% and stay connected to power through these steps.)
1. Unpair and reset Apple Watch. DO NOT SET UP AGAIN!
2. Back up iPhone.
3. Factory reset iPhone.
4. Set up iPhone while restoring from backup.
5. Wait for all your apps and everything to finish syncing and downloading from the restore. DO NOT PAIR WATCH YET!
6. Once your phone is done (it should still be connected to power) set up Apple Watch AS NEW. Do NOT restore from a backup.
7. Once Watch is set up, immediately go in to the Weather app and delete the pre-populated locations.
8. Tap the + icon and manually set your location on the watch. (Do NOT set this on your iPhone in the Watch app as it seems the weather data trying to sync is what causes this hangup.)
9. Enjoy longer battery life and hope that Apple fixes this issue sooner than later.
Since doing this, my Apple Watch Ultra (Gen 1) seems to maintain the same battery life it did before this issue started. From what I can tell, the Weather app is either trying to pull location data locally and gets hung up (whether you launch the app or not) or it is trying to pull data from your iPhone to sync your saved city lists/location/etc.
Also note that, even though you manually set your location, the complications seem to still have an issue populating the data set. You can tap the complication and launch the app and the data will be there, but it won't be on the complication itself reliably. This work around at least un-sticks whatever processes are broken in the Weather app that cause the battery drain.
This was my solution and I hope it works for you!