You would take the model of the CLI, start a temporary web server to capture the response.
Alternatively you use a Server or a webpage to handle the oAuth loop which then forwards the token to a customURI handler or some other way to relay to Unity.
The third alternative is an accounts system the user logs into your game using your accounts system, then on your accounts system you have a “link my Twitch account”. You’d need this approach if you were going to do drops for example.
The other alternative for bits:read is that you could connect to chat anonomously (using the justinfan login method, justfinfanX where X is a number and a no or a blank password) and then parse chat for bits/cheer events. So then you only need to ask the streamer for their channel name (and convert to lower case)