Using Google Play achievements is limiting in a number of ways but the biggest one is that the user may not be logged into Google Play. Another one is that this approach is limited to Android devices with Google Play installed so it won’t work for Amazon devices or iOS. I did just recently stumble upon something that might just meet your needs and cost you nothing (assuming you stay within limits). Google’s Firebase is free within limits and has a REST API for the Realtime Database. I found a Corona library from 2015 that is for the older Firebase DB REST API. I am not sure if it works for the current Realtime Database REST API. It might be worth checking it out though. Here are some links.
Newest Firebase DB Links
https://firebase.google.com/docs/database/
https://firebase.google.com/docs/database/rest/start
Firebase Pricing
https://firebase.google.com/pricing/
Old Firebase DB REST API
https://www.firebase.com/docs/rest/api/
Corona SDK library for old Firebase DB REST API
https://github.com/vsergeyev/corona-firebase
The Corona library is just a thin wrapper for the REST calls so it should (hopefully) be easy to adapt if it does not work as-is.