Hey Adam,
GeoREST supports (and is by far most performant with) the FDO SQLite format which is shared with OGR (though recent versions of OGR have added full Spatialite support). You can see details on this format here:
Unfortunately, the FDO provider for SQLite doesn't support Spatialite format natively.
There is a workaround, but it's a bit convoluted. If the SQLite file is created by FDO, Spatialite can bring it in as a virtual table and work with it as usual. I believe that you can use FDO Toolbox to create an FDO SQLite database.
Hope this helps. One of my personal desires is to see native Spatialite support in the FDO Sqlite provider, but it just hasn't happened yet.