Others see problems, I see solutions.
Problem the first: Japan wants to keep subsidising a dumb inefficient national whaling industry. As best I understand things, nobody really wants to eat whale meat and the industry would have just died. But, the rest of the world backed Japan into a corner where it would be embarrassing to stop doing it because they can't stop it without being seen to have lost this fight. So they keep subsidising it, the whaling boats keep going out, and everybody keeps going ahead with confrontational opposition methods that guarantee that Japan can't just cancel the whaling subsidies. And so a bunch of intelligent animals that would prefer not being killed continue to be killed for no particularly good purpose.
Problem the second: A bunch of intelligent animals keep attempting to commit suicide on New Zealand beaches. These whales seem pretty determined. Often, after their would-be rescuers float them out to sea, they just go and beach themselves again. They really seem to want to die.
In the first case, we undertake costly efforts to kill whales that want to live. In the second case, we undertake costly efforts to prevent whales that want to die from killing themselves.
And so here is my proposed solution.
BE IT RESOLVED THAT, henceforth, New Zealand will allow Japanese whaling research vessels access to suicidal New Zealand whales. For an annual fee sufficient to ensure that the Japanese whaling industry remains uneconomical and in need of continued Japanese subsidy, the Japanese whaling fleet can harpoon suicidal New Zealand whales, haul them out to sea where we can't see what's going on, then do whatever it is they do to them. If we haven't sufficient diversity in suicidal whale species, other countries with similarly suicidal whales may wish to join in.
IN EXCHANGE, Japan promises to cease all other whaling, since they'd have at least enough whales here to play with. The research vessels will also try to figure out why whales are killing themselves.
New Zealand and other countries get an end to objectionable forms of whaling. Japan gets to keep its whaling industry. Whales that want to live get to keep living. Whales that want to die are allowed to give their bodies up for science instead of winding up on beaches.
Sure, we'd need a few safeguards. For example, the maximum harvest shouldn't exceed what's currently taken to reduce the incentive to drive whales to suicide. And anything that looks like "let's do wonky things with sonar to make whales want to kill themselves" should reduce the quota limit.
Any reason this couldn't work?
Update: KiwiPollGuy raises some really good objections in comments; he knows far more about Japan than I do. I'll clarify things a bit. First off, I hardly think that "stupid" is the only reason behind whaling. I expect that it's an embedded cultural practice that has value for Japanese people even if they don't eat whale meat. The idea of seaside villages cutting up whale meat likely has the same romantic connotations for folks in Tokyo as the image of tiny Newfie fishing villages has for folks in Toronto, leading to public subsidy of lifestyles that are no longer economically sustainable. Canada was able to get away from some of this in the last twenty years - I understand that the 12/40 rule is now gone. But I've a hard time seeing Japan stopping the subsidies when everybody's yelling at them to end whaling. It might not be the only binding constraint, but it would be among them.
I am arguing that you could, in theory, maintain that whole basic deal by flipping the whaling ships from killing whales at sea to harvesting suicidal beaching whales. But it could be impracticable. I'd expect that beached whales go pong after a few days, and it might be impossible for whaling fleets to get there that quickly. Maybe this could be changed by having Hercules aircraft with air-mobile whalers and kit, ready to land at the nearest airport, truck over, and start to work while the Nisshin Maru trundles on over. We could even package it as a win for Japan rather than a concession - they get to flense whales with our permission in our waters and, depending on logistics, perhaps even on our beaches. I can't see how this is worse than a DoC guy shooting the whales instead. A more substantial constraint would be if the tasty kinds of whales aren't the kinds of whales that tend to go suicidal - if somebody who knows about whales could weigh in on that, I'd update. I know pilot whales keep killing themselves, but have no clue about their relative tastiness. I also don't know which other species have suicidal tendencies.
But KiwiPollGuy may be right that the NZ left would never go for it - that eating whales at all, even the suicidal and would-be-dead-anyway ones, is taken as evil. That would mean that even if NZ got behind a reduce-the-harm version of whaling, we'd get protesters on the beaches interfering with harvesting the suicidal whales. I can't see it working out without having the Greens on board.