It seems few people understand how the Kouhaku works. NHK sends teams of pollsters around the country from March until September. They ask people about artists, research what songs are requested at radio stations, look through Oricon's rankings, and then a committee selects a list to send invites to. some may accept, some may decline (in practice, only few decline), but it is NHK who decides who to invite, not the record companies or talent agencies. Can they try to muscle a bit? of course, but it all comes down to NHK sending the invites. They have been doing it for 57 years. And it has worked, in spite of recent audience drops, Kouhaku is still the most viewed music TV special on japanese TV year after year. What companies can do, as UFA has done before, is to put some other acts as back dancers for the act that was invited (they did that with MM, the kids were always there as back dancers)
EDIT: it also depends on the "theme" the Kouhaku will have that year, they ask "will this or that reflect the theme?". This year is 歌の力、歌の絆 Power of song, bond (or maybe feeling?) of song.