I apologise- it seems that it should be written as two separate words. Bare fist fighting is illegal, professional boxing without gloves, which is a clandestine sport often carried out on ships in international waters. I believe there is also unlicensed boxing, which is where the fighters don't have the authorisation of the boxing authorities to fight, but fight in much the same way as the normal boxers.
Ron's suggestion of 'brawling' would, in BE, refer not so much to the sport as to fighting in, say, the street, especially when alcohol is involved.
I'm afraid bare fist fighting\boxing is three words. When I'm at home, I'll try the OED and see if they have anything. ;-)