I have never heard of anything of that sort at Bain or BCG.
At A.T. Kearney, however, we have a large think tank that operates in a similar fashion to the McKinsey Global Institute. This think tank is called the Global Business Policy Council and is one of the most influential privately owned institutions of this sort in the US.
It is led by Paul Laudicina - former A.T. Kearney global managing partner - out of Washington, DC. Although I've never worked directly with them during my time at A.T. Kearney, I know they do solid and interesting work.
If you want to know more about the GBPC, go to: https://www.atkearney.com/gbpc/about-the-gbpc.