I decided to test The Weather Network since that's what I was always using and am most familiar with. Here's what I did in case anyone else wants to use it.
In private space I obtained The Weather Network from Aurora Store (I never realized it has so many trackers! 😅). The only permission I gave it was Network. I manually searched for my location as I don't trust this app to have it. I use breezy weather for my actual location. So yes, it works on GOS without Google play services. Just thought I'd share in case anyone else wants to use this app. :)