I’d do the auth on your website.
And your unity game connects to your website to collect the Twitch Auth to use, or the website proxies the events.
So for example,
Doom Eternal has a Bethesda login, which logs into Bethesda system.
You go to the Bethesda website, and you link your Twitch Account to your Bethesda account.
Doom checks the Bethesda server via the Bethesda login to check Twitch Prime rewards status.
So, your game logs into your server, and the server holds the connection data/account associations