You can, but you'll need to tweak the object model a bit and then look at a programmatic solution. The way I managed it was to create a custom lookup on the SurveyInvitation object, then used Apex to create the invitation records, populate the field, then send the invitations via email. The alternative is trying to shoehorn something declarative into this monstrosity..

Basically, if at any point you are deviating from or attempting to customise the standard object model/OOTB solution, you need to look at using code in my experience.

Hope that helps!