Regarding killing whales: legally, what you're describing is a very murky grey area (example US summary). You might be breaking international law, you might be breaking your own country's laws by doing this overseas, you might even be breaking a seldom-enforced law in the country where the tour is operated. For an example of what can happen when someone trusts, as you do, that the local organisers have ensured everything is okay legally and morally, and kills a much-loved animal overseas, see the case of Cecil the lion. It didn't go well for the hunter.

Regarding sportingly pursuing whales: there is a very good, 100% legal alternative, with no killing, harm or even distress to any whales, just a sporting hunt in the "find and seek" sense of the word, then peaceful mammal-to-mammal interaction. This looks like a good option since the asker mentioned they want:

Tradition, fun, thrill-seeking, and ... generally awesome ... Catch and release would be fine by me too

In Tonga (English-speaking) and I believe a few other countries, you can do tours which are like hunting whales, except when you catch up with the whales, instead of blasting them with an explosive harpoon which frankly can't miss because you're aiming at a frigging whale (in terms of challenge, surely the sporting equivalent of kicking a ball into the side of a barn), you jump out the boat with a pair of fins and a snorkel and swim with the whales. Mammal to mammal, minimal technology.

If that doesn't sound awesome, check out the final scene from the "Mammals" episode of BBC's Planet Earth series, where you see the boat crew and underwater camera crew chasing humpback whales in Tonga. I can't currently watch video but I think the appropriate clips are on this page (feel free to edit in a better link).

If it's tradition you want, I'd firmly recommend the way I did this, which is from Eueiki Island in Tonga. The owner of this private island has a motorised canoe based on the traditional design that the first Polynesians used to colonise this part of the South Pacific - those ancient Polynesians knew what they were doing, and it's faster and quieter than the other whale boats around (as well as very stylish). He also has 100% traditional sailing canoes and is an expert sailor, but it'd take a long long time to catch a whale the 100% traditional way.

You'll go with a Tongan navigator who knows all the traditional tricks for reading the seas and working out where a whale may pop up next. It's very difficult, and it makes for great sport. (certainly, a more honest sporting challenge than using radar then blasting one of the world's biggest mammals with an exploding harpoon)

Then you have to be ready to jump in the open sea at a moment's notice when you finally catch up with a whale. Usually there will be two or three whales together (mother, calf and male escort is pretty common), and, if you can keep up with them, you'll get an unforgettable half hour or so diving underwater, watching them in their natural habitat, swimming up to the surface as they do to see them breach, etc. Whale calves are quite curious too, you might get a little unforgettable interaction as a van-sized baby whale swims up to take a closer look at you.

If you do want to kill things too... Well, that's still an option. The owner of the island is a keen spear-fisher, and has a dog who's trained to bark when suitable fish swim close to shore. There's excellent snorkelling all around the island, and if you can convince him that you know what you're doing with a spear-fishing gun, I'm sure he'll point you in the right direction to hunt your supper. (not whales though, that's very very illegal)

Regarding ethics, some organisations speculate that whale-swimming trips might alter whale behaviour, and so in Tonga, to protect the whales, there's a law that you can't go too close to the whales. It was 30m when I went, might have changed - but in those crystal-clear waters, that's easily close enough, and frankly, going closer than this unless you're a trained professional would be very dangerous anyway.