There is nothing that can be done about that "automatically". The Google Play Store license works only through Google Play services.
That "Purchasing" page describes the alternative licensing mechanism, but you cannot just simply transfer your license from one to another one.
I don't know, - may be the developer, Kostya, - would be willing do help you out. I'd suggest sending him an e-mail to the address listed on that page. Note that Kostya is away for a week, and he will be able to read your message only after he is back.