I was facing the same issue a few days ago.
So I built a library that solves it:
https://github.com/KonstantinSchubert/request_data_webviewclient
It is a WebViewClient with a custom WebResourceRequest that contains the POST/PUT/… payload of XMLHttpRequest requests.
It only works for these though – not for forms and other kind of request sources.
The hack works, basically, by injecting a script into the HTML that intercepts XMLHttpRequest calls. It records the post/put/… content and sends it to an android.webkit.JavascriptInterface. There, the request is stashed until the shouldInterceptRequest method is called by Android …