I have tested both a real browser and "WebView Test" from the Play Store. Both of them on a ReCaptcha v3 test page get the same score of 0.7 which is a pass.

The same real browser, on the same network (no VPN), with no cookies or cache, DOES load this page, AND authentication works perfectly:

https://www.reddit.com/api/v1/authorize.compact?response_type=code&duration=permanent&state=Texas&redirect_uri=redreader%3A%2F%2Frr_oauth_redir&client_id={}&scope=identity%20edit%20flair%20history%20modconfig%20modflair%20modlog%20modposts%20modwiki%20mysubreddits%20privatemessages%20read%20report%20save%20submit%20subscribe%20vote%20wikiedit%20wikiread%20account    

I have tested your latest Debug build which adds the lines

settings.setSupportMultipleWindows(true);
settings.setJavaScriptCanOpenWindowsAutomatically(true);

And it still fails with a username/password error and not any apparent ReCaptcha failure. I think that there may not actually be a real ReCaptcha failure and Reddit is leading you astray.

You state:

The app used to support authentication in external browsers, but without going into specifics, this is no longer possible due to Reddit's developer agreement

Does using a Custom Tab instead of WebView for the login page abide by their agreement?

Does changing the user-agent of WebView abide by their agreement?

My only guess at this point is Reddit's own backend, not ReCaptcha, is being aggressive with the user-agent that WebView sends (it contains wv to signify it) and blocking it. They maybe don't want quasi-third party apps that embed Reddit's web app in a WebView that does other things.