i am unable to install the latest update on my series 3 watch...I seem to be caught in a loop.

Try the second option in there - updating the watch directly without the iPhone.

Update directly on your Apple Watch

If your Apple Watch has watchOS 6 or later, you can install subsequent updates without your iPhone:

1. Make sure that your watch is connected to Wi-Fi.
2. On your watch, open the Settings app.
3. Tap General > Software Update.
4. Tap Install if a software update is available, then follow the onscreen instructions.

Leave your Apple Watch on its charger while the update completes. Don't restart your Apple Watch, and don't quit the Watch app. When the update completes, your Apple Watch restarts on its own.

If you run into storage problem, check these additional instructions that overcome that issue using two unpair & re-pair cycles.

If you don’t have enough space to update your Apple Watch - Apple Support

Read HERE

Or;

The apple watch 3 GPS only has only 8GB of internal storage, most of that is taken up by the operating system and other critical software. So installing a major update is not going to be easy;

1.Unpair and erase your watch to factory settings.

2.Set up the watch again,but setup as new not from a backup.

3.Update the watch, which now has enough free memory to update

4.Unpair and erase the watch a second time

5.Re-pair the watch with your phone, but choose the backup you made in step 1,another backup is made in step 4 but don't choose that one.