I don't know if this helps at all, but I'll include it regardless.
It will help some day for sure. But for now, this plugin only relies on webpages that have the lyrics provided in plain text.
The synchronized versions of the Musixmatch lyrics are not available for viewing on any webpage (just in web-based applications like your Spotify, Instagram, etc.)
The original author of this plugin had structured it in a way that catered for various retrieval methods - static, search, and API.
At the moment, only the first two are implemented. I assume the "token" game you mentioned that other plugins are using is API based?
That would all be new territory for me I'm afraid. I would have to research a few things for which I unfortunately no longer have the kind of time for nowadays.
But yeah, maybe some day, one of us here will be able to get the API side of searching on this plugin going.
By the way, on the forum we have somebody who's managed to create Genius and Museexmatch plugins that have direct access.
So you may try contacting slonopot to see if he wouldn't be able to fetch synced lyrics on his side:https://getmusicbee.com/forum/index.php?topic=37508.0