sqlite_rsync can only be used in WAL mode. A further constraint of WAL mode is the database file must be stored on local disk. Clearly, you'd want to do this almost all the time, but for the times this is not possible this utility won't work.
If you are able, please try out this enhancement and let me know if it solves your problem. See <https://sqlite.org/src/info/2025-05-01T16:07Z> for the patch.
So, sure, from a low-level point of view, the write transactions are not really simultaneous (they never are with SQLite3), but from the user perspective they are in the sense that you don't need to serialize them yourself.