I would go with the special grease. I am not usually brand loyal for the sake of brand but other times a product stands out as a better match for specific applications.
I googled the grease and it looks like it will stay put with little to no bleed. Maybe other brands would work with similar characteristics like a TRC paragon grease iirc. It might be an alternative. I've read its suppose to have little to no bleed and has a high Timken load test, stable workability, and resistant to wash out. It also won some industrial maintenance awards but doesn't mention being synthetic so for compatibility I don't know.
In this application I bet a little specialized grease will hold up better than retail grease. Any ole retail grease might work but require more frequent lube and might contaminate other lube.
My bet is GM found this grease to fix the clunk better and it won the bid so to speak (after the fact not original design). Similar to what happened with Castrol Syntorq in the NV4500. And how in these applications manufacturers get to keep their trade names involved.