3 hours in the background seems a lot to me.
I started the app clean and configured it from the start (it was a pain in the ass because I migrated my settings from Sync for Reddit lol) and the issue still remains, having this huge background usage prevents my phone from going to deep sleep properly which lends to a bigger battery drain.
EDIT: BTW the issue is gone when I set the app to restricted battery mode instead of optimized... But I'd rather keep it in optimized as it closes less often in the background, that is how I had it with Sync for Reddit, which on average its active background was several minutes only.
EDIT 2:
This is the process that Battery Guru/BBS says shows
job/io.syncapps.lemmy_sync/androidx.work.impl.background.systemjob.SystemJobService#9188
So I assume you experience the same issue?
I see Sync spending about half its total time in the background. I'm also curious why.
I have my suspects that is because it tries to fetch new messages through a non existent push notifications feature/API that was transferred from Sync for Reddit... But hell, that is just a non funded suspicion.