Android seems to have an issue with the PATCH Http verb, which is necessary for Easy Tables' Update. Investigating
Unity has acknowledged that this is a bug, waiting for a hotfix
Unity has responded that this is by design, due to Android http client implementation that forbids PATCH.
Investigating potential other workarounds (apart from doing an Easy API call on Android just for Updates)
Azure team response: http://stackoverflow.com/questions/39023937/trying-to-use-patch-on-a-unitywebrequest-on-android-and-getting-unsupported-pro
Unity team response: http://answers.unity3d.com/questions/1230067/trying-to-use-patch-on-a-unitywebrequest-on-androi.html
So, apparently PATCH is not going to work with Unity on Android. Implemented a workaround which requires EASY APIs, check these two gists: https://gist.github.com/dgkanatsios/b380d1ae4195b6592c1337c63c56b2ea
This is how the Mobile Services SDK for Android handles PATCH - wonder if Unity could do the same workaround for UnityWebRequest...?
The Mobile Services team opted to use the Apache HttpClient for Android instead of the native Android Http client, probably for this very reason. Unity could potentially use that, but I think that this would require some refactoring on their side.