I don't know if Japan is really short of heavy helicopters for dumping water. I saw a news clip that showed a Chinook doing a water drop. That's about all that the U.S. could send over. Sure, Russia has some heavier helicopters (and technically the Jolly Green or the Sky Crane could be bigger but there's not many of those around any more) but I'm pretty sure Japan is using as heavy a helicopter as they've got.
And you're probably partly correct in that a fair bit of help isn't provided simply because Japan hasn't asked for it. I think that would make them "lose face" by admitting they can't handle their own problems.
