Ballistol does work really good. I started using it about a year ago, and it gets the fouling out really quick. I used soap and water for over 20 years, with Ballistol the patches always come out squeaky clean on the rinse. I use two buckets, one for cleaning and one for rinsing.
I have no experience with 777, but for regular BP, Ballistol works great. The only problem with it is that it stinks to high hell!