I’ve been looking into this solution again, so I just thought I would provide an update that might be useful for future plugin development.
A few plugins, such as WP Migrate, handle database table migration quite well.
However, WP Migrate will not let me sync specific posts tables without removing all the other non-selected posts tables. I believe this is because they want to avoid doing any kind of database merging, which is explained here.
If L&L wanted to simplify the migration of templates to other websites/domains, the team could implement something similar to the Local JSON feature for ACF:
With this enabled, it becomes a simple case of copying the .json files between websites/domains and can be automated using whatever file copy process the user prefers. 