It seems you don't want to keep the whole DataTable as a copy, because you only need some rows, right? If you got a creteria you can specify with a select on the table, you could copy just those rows to an extra backup array of DataRow like
DataRow[] rows = sourceTable.Select("searchColumn = value");
The .Select() function got several options and this one e.g. can be read as a SQL
SELECT * FROM sourceTable WHERE searchColumn = value;
Then you can import the rows you want as described above.
targetTable.ImportRows(rows[n])
...for any valid n you like, but the columns need to be the same in each table.
Some things you should know about ImportRow is that there will be errors during runtime when using primary keys!
First I wanted to check whether a row already existed which also failed due to a missing primary key, but then the check always failed. In the end I decided to clear the existing rows completely and import the rows I wanted again.
The second issue did help to understand what happens. The way I'm using the import function is to duplicate rows with an exchanged entry in one column. I realized that it always changed and it still was a reference to the row in the array. I first had to import the original and then change the entry I wanted.
The reference also explains the primary key errors that appeared when I first tried to import the row as it really was doubled up.