I'm not sure I totally understand your situation - but a couple of approaches come to mind:
1) Use the report query to do the update. So you would take the original Processed data set and stuff it into a temp table; then issue an update for the Completed flag; then select your data set for the report back from the temp table. I think this can be done. It's like you're issuing a series of SQL statements as you would in a strored proc.
2) Use the SSRS subscription to output the report to, say, a PDF instead of email. Then use Database mail to send the PDF as an attachment. Then update your Completed flag.