Firstly, acknowledge that in this model your schema IS your API. Don't include anything in the database that is being replicated that isn't considered to be documented, tested and stable.
With that in place, API changes become the same as changes to a JSON API or similar: you can only make backwards-compatible changes, like adding new columns or adding new tables.
If you do need to make a backwards incompatible change there are ways to patch over it. Ship a brand new table with the new design, and include a SQL view that matches the name of the old table and exposes a compatible subset of it.