How to export a growing Aperture Smart Album repeatedly without creating duplicate files
One method you might consider is using Aperture itself to create your web gallery. Use File > New > Web Page to create a web album, then fill it with photos from your smart album. Aperture won't duplicate photos in this album even when you drag in duplicates, so you'll end up with one copy of each image after all of your updates.

When you publish the web album (using the Export Web Pages button), Aperture will replace any folder with the same name, again avoiding duplicates. This perhaps isn't quite as good as just skipping duplicates -- it does waste time, deleting old files and recreating identical new ones -- but it nevertheless achieves your goal.



But if you want to stick with your current system, the bad news is that there is nothing that will force Aperture to directly overwrite existing data. It's one of the fail-safe mechanisms of the program. There is a thread on the Apple discussion forum on a nearly identical question, and one of the respondents there suggests creating an Automator action that will delete the existing images on your Dropbox, and then export a new set of images from Aperture. Again, not ideal, but it does achieve your goal.