On average, Legend of the Streets give you more points. You get
more only on the higher level tracks and com drivers with higher
skills (last 2 or last char for the track). Points gotten ranges
from about 7k - 11k. Also if you challenge them again on rain you
also get, I think, 1000 more points than the dry challenge. This
only happens once so if you challenge those 'bosses' again, you'll
get just 3k - 4k. Challenging all the weaker coms and challenging
the same 'bosses' more than twice give you only about 3k - 4k which
is about the same as Time Attack.
Time Attack on average gives you 3k - 4k unless you can break your
previous record in which case you get 1000 bonus.
Battling won't give much if you count in both players. Loser always
gets 1000 and winner will get about 4k - 5k. Unless if you can
continually win, then you get the straight win bonus.
Bunta will only challenge you on Akina not Iro. Bunta is still the
last one to challenge you and it is free only for the first time he
challenges you. No short cut, you need to unlock all the other
'secret' drivers first on the other maps before Bunta appears.