A blessing or a curse?

You pretty much just said it. Find a Wailmer nest to camp out at or get good with your Regice throws. Regice actually isn't terribly hard to get excellents on, with some practice you could hit an excellent every 3/4 throws at least.

I've also found stantler and slowpoke to be really easy.

As you noted, get the first one on a harder Pokemon before going for the big hit box, so you only need 2 under pressure.

Try to use pokeballs so the Pokemon hopefully breaks out and you can get the feel on the same Pokemon. Nanabs as you noted would help too.

Finally, make absolutely certain you're throwing for excellent. Worst would be to hit a great.

Honestly, your best bet for this is still Regice. The biggest problem with this quest is "3 in a row", and most Pokémon you're going to catch it on the first or second - and if you don't, it's going to flee and you need to find another Pokémon with a good Excellent Spot. So you need a Pokémon that is HARD to catch - Regice is about the only one that fits the bill. The best thing about Regice is that the Excellent spot is the same at all times, unlike Ho-Oh, Lugia and other Legendary's that move up and down on the screen. The second best thing about Regice is that he can be extremely hard to catch - I hit it with 7 straight Excellent Curveballs with GRB's in one raid, and still didn't catch it. (Yeah, I'm pretty good at the curveball, but I'm not THAT good - Regice just happens to be right in the middle of my sweet spot.)

Other Raid Bosses I've had luck hitting Excellents with are Machamp, Absol and Tyranitar - though I don't recommend Ttar because of how much of a (bleep) it is to hit, unless you're good at timing its attacks. I'm not particularly good at timing Ttar's attacks, personally - I can hit the Excellent Spot, but half of the time it attacks before the ball gets there because my timing is terrible.

If that isn't an option, these are the ones I've had the most luck with:
Spheal
Wailmer
Hoothoot
Doduo
Ledyba
Kakuna
Flying Evolved Forms (Venomoth, Pidgeotto, Pidgeot, Butterfree)

(edit: Forgot Slowbro. Yeah, that's a one for this too. Big Excellent spot, not much in the way of attacking or dodging. Only problem with it is that it's really easy to catch, so you can't get 3 in a row on it. It's great for the 3rd one, though.)

But I'd still recommend a Regice Raid, on a Gym that your team controls. Once you figure out how to consistently hit its Excellent spot, cross your fingers for a low IV Regice then don't use any Berries (except Nanab or Pinap) and just complete the quests.

I've been hoarding wailmer and slowpoke quest rewards in my stack since the start of the water event. Now I just get the first excellent on a hard Pokemon, and the next two from my stash. But as others have said, the iceberg is not difficult to hit excellent on, and he is a good target to practice, sine you don't really care if you catch it or not.

Is articuno worth more than larvitar? I'd delete one of them and get the stamp.

I usually find a nest with a million of the same pokemon, practice to get my timing/aim standardized, and it generally works out pretty quick.

As others have said, Regice would be the way to go. It doesn't require much of a curve or that strong a throw. It's simple punch doesn't require that much of a windup after you've set your circle. Plus, it doesn't move around allowing you to get distracted. While I haven't had a larvitar quest at stake while trying to catch a Regice, in the few that I've done, I've had long strings of excellents in a row (since its catch rate is pretty low, it tends to jump out of everything).

In light of wailmers being rare again, another wild pokemon that you might have around you that I've found to be useful for this quest is Nosepass.