We have deployed this, but I haven't had a chance to test it end-to-end with Survey Monkey. I know that it will work, provided that the integration is as you've described.
If you have your Optimal Workshop survey set up, and your Survey Monkey survey set up, could you please run through the end-to-end process yourself once or twice and make sure that the participant identifier is carried over successfully from Survey Monkey into your Optimal Workshop results? I am fairly confident that it will work, but would hate for you to invite your participants to the survey and have their time and yours wasted. Let me know how you go.
Depending on what tool you are using, the identifier will show up in a different place in your results. In Chalkmark, you'll find them in the participant list download (the spreadsheet on the Downloads tab), listed as 'login entry'.