It sounds like what you really want is a multi-master/bi-directional replication.
SQLite isn't a fully featured SQL database so it doesn't have this built in by default. Other SQL engines, such as MySQL, SQL Server, PostgreSQL, etc. all have this feature. There are SQLite servers that will make SQLite addressable from multiple machines at the same time, over their own network protocol, but thats just a poormans version of a proper SQL server and you'd need to modify lapper to support it anyway.
If you wanted to do it automatically every X hours you'd have to roll your own script to merge multiple SQLite databases - it wouldn't be too hard - you just need to give each entry its own unique id and the last change time of each row, and then check each row that doesn't match in each database and update where necessary (most recent update wins - but you'd need to always favour one in the event that both get updated at the exact same time).
I've not looked at the databases you've provided, but in theory it's possible. You'd most likely need to make some alterations to your SQLite database - either LFSLapper would need to update a column recording the datetime of the row change, or you'd need to add a trigger to each table that does this instead.
I don't have time to produce something that would do this, although once you've got the datetime and primary keys in the database it's actually fairly trivial as you can make the script or program intelligently able to cope with any table by utilising the describe statement, but hopefully this will help you/someone along with the process.
Edit: I've just realised I might've gone off on a tangent and not actually answered your question - if thats the case then I apologise.