I found this thread trying to figure out why my own BrowserSource overlays weren't working - and thought I would take a look at the code in this plugin.
It has the same issue I was having - this plugin uses BroadcastChannels to communicate between an OBS Dock and a BrowserSource on a scene, however OBS27.2 has changed the way it technically implements BrowserSource elements if you select "local file"
Docks are added (as I would expect) using the file:/// scheme - (eg file:///c:/obs/file.html)
In previous versions of OBS, BrowserSource local files were added the same way - nice feature, BroadcastChannel works to communicate between the two windows
In OBS 27.0 - BrowserSources set to Local File add the file using an absolute/ scheme (non-standard afaik) (eg absolute/c:/obs/file.html)
This breaks BroadcastChannels - as they only work using the same scheme and host.
FIX / TLDR:
Change your BrowserSource from a Local Source to a manually entered file:/// uri, changed all windows back slashes in the file path (\) to standard uri forward slashes (/), and replaced any spaces with the uri encoded %20 for good measure.
eg Local Source c:\obs project\file.html becomes standard url file:///c:/obs%20project/file.html