I've been through this exact scenario for a government website a short while ago, they also wanted an exit survey.
Because of the futility of trying to get a "nice" implementation of this and many usability and accessibility concerns that I pointed out to them.
We ended up using a "survey" button that's anchored to the left side of the screen and a call to action on the home page for the first month the survey is online. This button would in turn open a modal window (and degrade gracefully by going to a separate survey page when javascript is disabled)
I can see if I can find an example if you're not sure of what I mean

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