Having not used this product, I think the official install is a bad fit for this environment.
This appears to be for running simple independent containers, which works well if you have all your data on a file share or separate database. In the case of the standard Discourse install, the container is the app, the database files live on the local disk, and the container management is a script called launcher.
For Synology, it seems you’d want a way to build images on another system, arrange for the the /share/ mount to be on a file server somewhere, and replace launcher with the Synology controls. Doable, but not a simple drop in.
Or run containers inside containers.