You can’t
A twitch extension is designed to be used one place and one place only. On Twitch.
To render the extension information on another site that extension would need to provide an API to consume or it’s own non Twitch version of the extension for embedding.
Please consult with Streamabs for this use case, of which we are not the support for, or build you own with which we can help with.
TLDR: you can’t do that