Hi Richard,
I understand your requirements to let customers download the pictures easily. However, the function to download a copy of the file, like the Dropbox does, can only be fulfilled in the library, and libraries on SharePoint public website cannot be accessed anonymously for the time being.
Since the Open button will not show up, you can try the second solution to check if it can work around this problem. The following article is also referring to downloading files on public website, and also mention that the images will open in the browser, but the customers won’t see a prompt to download the file:
Sorry that your requirement cannot be fulfilled perfectly. However, your suggestions and comments are very helpful for us improving our services. I'd like to invite you to submit feedback to us about your concern and advice via the following link:
Our product team will always collect the feedback from our customers and make the product better. Thanks for your time and efforts.
Best regards,
Nancy Xie