I originally posted this question in the issue queue for Background Images, but I'm cross-posting here as well just in case it can be supported in this module instead. Thanks for your consideration.
Nope, background css images are not clickable.
Automatically closed - issue fixed for 2 weeks with no activity.
It turns out that miracles sometimes do come true, thanks to @malcomio: https://www.drupal.org/node/2424675 (#8)
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