Hogosha are treated as lawful a faction in Kusari. Following your rule, we can't fufill GMG bounties as well, but there are (miraciously) registered BHG. Same with Council (unlawful ID but they are treated as lawfuls by role) and Kusari Exiles (same status within all four Sirian houses).
The very same problem may and WILL apply to The Core, who are to become quasi-lawful faction.
As for "lawful bounties", in they are bounties that does NOT target lawful targets, unless houses are at war. Angels replied on that question like this. In short, we can claim bounties from Hogosha except for bounties targeting corporate and military factions.
Also, this gives out a huge hole in the rules. If we made your approarch, while escorting IMG, BHG are not allowed to help IMG if they are pirated by Kruger. Just saying.