I don't think you can add a button to refresh the extract (although if someone else knows of a way, I would love to hear about it!). However, I think it should be possible to manually refresh the extract with a new set of parameters without having to download the workbook. This is of course provided the workbook has been uploaded to Tableau Server and the users have the appropriate permissions.
First change the parameters in the workbook on Tableau Server. Then go back to the workbook's list of views. Before selecting a view, near the top of the browser window next to the name of the workbook select "..." (More Actions), then select "Refresh Extracts...".
I have never tried this, but to the best of my knowledge I think it should work. I hope that helps!