Most of times PHP is used as a script from a web server, so the script is just run and closed. There is no time for database synchronization as the script will close before the data is transferred.
We can only use LiteSync on PHP when the script is run as a program, and it is kept open for a while.
The synchronization is not slow, but it needs time for TCP communication with its peer.
Even on these cases we need to patch PHP to accept URIs for SQLite database names. The PHP engine blocks these names and it is the reason for the error message you shared.
I can see that you are using PHP Desktop. It uses a web server in the background (Mongoose). So in your case there is not enough time for synchronization.
What you can do is to keep a PHP (or even other language) app running in the background serving database requests, acting as a small database server, to which your PHP scripts can connect to to request data.
But in this case a MySQL server would do a better job. This is the reason why PHP is not listed on the supported languages. Although it can be used with Laravel.