"Export-DataTableToSQLiteDB : DataTable already belongs to another DataSet."

It appears that the solution was to go away for a little while and then come back with a fresh mindset.

Upon returning I had a "Duh" moment and created a new datatable with the same columns as the source one, copied the rows into the new datatable and then I had one that didn't "already belong to a dataset". 

Hey Matt, glad you got it working.
A simple solution is to just copy the whole data table. Something like this:

foreach ($table in $Dataset.Tables)
{
    $table_copy = $table.Copy()
    Do-Whatever -InputTable $table_copy
}

Thanks, Zachary. That's even easier. 

Thanks, Zachary. That's even easier.