A proxy server may be a better option here.
As to be selfcontained inside OBS, you’ll use implicit auth and that token is non refreshable and valid for only 60 days, sure yo ucan store it in storage between restarts.
But when the token dies the streamer will likely need to interact with the browser source to grant a new token. Which might be sometimes seemless but difficult if it breaks for a user to understand a fault has occured and might need to manually perform a task. (Since you are also in a self contained context without access to normal password managers/etc)
Websocket subs only exist for the lifetime of the socket.
Webhook subs stick forever (until clientID revoke)
My personal choice would be the proxy server here.
Or run a desktop app that handles authentication and serving of the overlay, as if the token dies I can present a suitable error and reauth inside the app. And said app can also act as an event reader/list which is easier to scroll to see history/etc.