TLS 1.1, 1.2 in WebView for android <= 4.3

I'm not sure if you can make it work with the current WebView but I would recommend you to have a look at: https://crosswalk-project.org/documentation/android.html to replace the WebView.

This is built upon latest version of Chrome and you won't have problems with that version of Android the user is running on.

I'm sure there are other alternatives too, but this is the one I'm using for work and I know work.

Best of luck to you.