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At this stage of things, I would try and contact your bank and open up a dispute and inform them what happened and that the developer will not give you a refund and let the bank secure your refund from them.
The problem you will run into with the bank when it comes to Google Play is that they generally will take their side that the vendor/software developer should pay. That leaves you the option of filing a small claims demand against Google but due to the value of $7.97, it may not be worth your time and effort.
I would write a review of this app and warn people from making any future purchase based on your experience.
Also you can make a Federal Trade Commission complaint against Google here, that way if enough people have similar problems, you may be entitled to a refund from Google - See: