There is one way to get your series 3 apple watch to 8.7.1 update.
I successfully did it with my wife and my own series 3 apple watches
Pre-requisites
a) Having iphone 8 or newer, you can do it with your friends iphone just for the update, you can unpair it later( I used iphones 8, 11, and 12 models, I did not try olders).
b) iOS 15.4.1, it won't work with iOS 16 no matter how many times you try.
c) apple watch series 3 in fresh factory reset state.
Steps
1- turn updates OFF in apple watch and iPhone (both charging and WiFI)
2- delete all apps then can be deleted from apple watch, ALL.
3- pair apple watch with the iOS 15.4.1 iphone and let the process be completed
4- from apple watch go to software update and wait long time until the update is downloaded
5- go to settings, general, and tap install in apple watch (or in iphone, both of them work)
6- wait very long time (2-3 hrs) until process is completed
7- enjoy watchOS 8.7.1 fix for series 3
If you did this with somebody else's iphone, just unpair it then pair it with yours.
this works for both series 3 apple watch sizes