It may be more of a drastic change, but have you considered moving to a backend that’s built for concurrent access? PostgreSQL is an excellent, open-source and free option, and we use it to manage thousands of concurrent clients across our servers.
I’m not sure why running the PRAGMA command resets the connection. The Omnis docs indicate the SQLite connection runs with the PERSIST journal mode: https://developer.omnis.net/onlinedocs/Programming/09serv.html#troubleshooting
The page also suggests the mode can be changed to OFF, so I would expect you could set it to WAL. Perhaps email Omnis support to ask about this?