Default rewards, like “highlight message” and “unlock emote” are not “custom”. So yeah they are not reported as a Custom Reward (nor are their redemptions)
You’d need a way for the widget to ask for oAuth and be able to write to the storage system SE uses.
Like I mentioned trying to use SE’s system is crippling you.
Without a server to handle oAuth, you are going “round the houses” to create a round about way to achieve the end result.
In this example:
https://barrycarlyon.github.io/twitch_misc/authentication/implicit_auth/
You can login via Twitch, and the page will use the returned implicit auth token, but it doesn’t try to store it anywhere else.
Concievable your SE widget could say “I have no token please click here”, then the broadcaster has to (in OBS) interact with the source,
click the link
login with Twitch
2fa with Twitch
then the SE Widget has the token
Then the SE widget could write the token to the SE Storage (in theory if thats supported to write to the secure storage from a “public” widget which sounds dangerous to me)
But, the token could die mid stream, then the streamer has to do all this faff whilst live,
and run the risk of accidentally leaking their password (if they don’t hide the widget and/or end up typing in the wrong field for example).
It’s a very not optimal solution.