You can look into github LFS. It’s fairly cheap as long as you never change any of the large files. Not sure if that’s possible for you or even with Lumberyard.
We are uploading just the project directory as we’re mostly working on scripting. We’ve had some issues with it but it kind of works pretty often.
There doesn’t seem to be a good solution. I recommend keeping backup saves on a local machine and trying to store only the important scripting stuff in github.