I have had this problem. Essentially, precompiled binaries (SQLite3.dll,
and System.Data.Sqlite) for Windows do not appear to allow extension
loading using the SQL function.
I have raised this before,
http://sqlite.1065341.n5.nabble.com/Extension-Loading-td93885.html but it
may need custom compiling although I'm not sure what flags would be
The Windows CLI does allow loading by SQL function
I didn't find a resolution here or at Spatialite