My understanding is that just "using Realm Object Server" means using Realm Platform Self Hosted, which starts at $2000/year. The hosted Realm Platform starts at $30 month, but I've read you need to upgrade to a similarly expensive $1500-2000 enterprise plan if you want to access the data from non-mobile-Realm clients. That is to say, I'm very open to just using Realm Database on the client for local persistence, because it's free and really nice to use, but don't think Realm for sync is viable.
AWS AppSync looks like it's worth investigating, thanks!
Agreed a custom data layer/cache might be the way to go, I think it gets a bit more complex once you have a bunch of objects being edited and deleted, and I've also spent some time with iOS background networking in the past and I know it can be buggy/frustratingly non-deterministic, which is why I'm reluctant.