Immich has an app that just needs to connect to an Immich server, which this would effectively be. It does the background sync, though I don't think it does the delete part.
This was a few years ago, and was one of several "iOS breaks our crucial workflows" problems that motivated switching our last device over to Android, so the situation might be better now...but I've never had that problem with the Android client, so I know it's not an Immich problem.