There is no way to know the base HREF until you upload to Twitch.
So you’ll either need to find how the framework operates without a base.
Create a shim to dynamically determine the base
Try <base href="./"> as the base which I have no idea if it works or not as a valid value.
The framework really shouldn’t need a base as
A document’s used base URL can be accessed by scripts with
Node.baseURI. If the document has no<base>elements, thenbaseURIdefaults tolocation.href.
As MDN states there is a backup/default to use instead