If Cologne doesn't want a war, why did she kill
someone? If she's going to kill one of them, why not kill
all three? It's highly unlikely that she's their only
stop, so the Yakuza would have no clear idea who killed
their men. Kill the next group, without giving them a
clear idea who, and unless the oyabun is a lunatic (and
they rarely are lunatics) they'll just go look for more
vulnerable prey. That's how organized crime establishes it's
boundaries in the first place. While it's likely that
this oyabun is a lunatic given his insistence on moving
in despite Nerima being a charnel house, Cologne doesn't
know that because she hasn't been ther that long.
Apart from that, 5 deaths a year in a Japanese
suburb, is enough of a lethal crime rate that letting
the Yakuza take over would be generally an improvement.
The police would _not_ be happy to see bodies being fished
out of the canal several times a year even if they were
Yakuza bodies and would likely assume assume that there
is a nasty crime family based out of Nerima. If the
Yakuza were to move in, then being aware
of this terrible death toll would probably be enough to
convince them that they should just let the fast food
place with the ninja slide.
Also, Nerima is hardly crime free if you consider all
the brawls, the property damage, the sexual assaults,
the underwear thefts, breaking and entering et cetera
and so forth. It's just of course that nobody reports
these crimes.