Contact syncing is for your phone to sync your contacts with Google. The contacts app on your watch is the companion app to the contacts app on your phone. As such, when your watch and phone are paired, your contacts are just on your watch, period; and they will only update if you add or delete a contact on either device, and then only for the period of time it takes to add/remove the contact(s) - less than a few seconds. If your battery is draining abnormally fast, something else is causing it. You may have some apps running in the background. Check your open apps on the watch by pulling up from the bottom and tapping the recent apps icon. Then close everything. You will be left with a screen showing background app (usually Samsung Health & Google Assistant). You can also check your battery usage in the battery section of the watch settings to see which apps are using your battery- I assure you it is not contacts. You might have to do some maintenance on the watch such as deleting the cache partition.
