Try just something like:
DateString:=Query.FieldByName('DATE').AsString;
But as Lacak pointed out, your DATE isn't a properly typed DATE at all inSqlite ...It is a string type.
Get it out of your head it is typed DATE. It isn't. It's a string.
And your format is also incompatible with the Sqlite date conversion functions.
If you use a properly typed DATE, be it TEXT as ISO8601 strings ("YYYY-MM-DD HH:MM:SS.SSS"), REAL or INTEGER then you have a properly typed DATE which is easy to convert to a
representional format through FormatDateTime combined with Field.AsDateTime.