WR

This is faaaaaar and above the most RNG heavy run in the game. Grinding for this took over 2 weeks and hundreds of attempts.
The estimated RNG for a perfect 4-6 run is 0.0135% or a 1 in 7394 (calculation is at the end of the description)
EDIT: It is worse. I forget a 60% roll. The actual chance is 0.008% (1 in 12323)

A lot has gone into making this as fast and consistent as possible:
First up you may have noticed that I don't have QA glitch active. Reason for that is that I am not playing on 1.9, which saves time later.

Mario's position in 4-6-1 always causes a fast move cycle for the Valkyrie and has the enemies positioned to maximize the chance of the burn crit leading Spawny into the goal zone.

Between battles there is a small puzzle. On 1.0 if you press the buttons twice you can time your second button press, so you are able to move while the platforms are moving. This saves around 8 seconds.

4-6-2 is interesting. The idea is to manipulate the enemies to be in a good position in turn 2 to use them to bounce Spawny across the gap, with Luigi's Sentry.
Four conditions need to be met to be able to do that:
An enemy has to be in a suitable position to use the Sentry on
Luigi needs to be alive (Use Extra Health!)
Spawny need to have enough health to survive the Sentry (EH!)
Spawny needs to be in range of a suitable position to get launched (with the help of Itchy Feet if necessary)
The first condition is usually given.
Luigi's position in the vid is the most consistent one I found. He is behind half cover meaning the chance of the Hoppers burning and causing chaos is even less likely.
To make sure Spawny is healthy enough to survive the Sentry you can bring Rabbid Peach with Heal as backup.
For Spawny's position condition we again make use of 1.0 again. It turns out out of bounds works differently on here than 1.9. Notably if you bounce Spawny from the position shown in the run he will make it to the other side. On 1.9 you would need Spawny to be one tile closer to the goal. This is very significant, since the position for Spawny in turn 1 can (with maxed out Itchy Feet) just barely reach far enough to get bounced across on 1.0, but not on 1.9. This makes 1.0 way more consistent as you don't have to rely on enemies to bring Spawny closer. But it gets even better. In my run I got the best possible RNG for 4-6-2 which has a chance of only 4%. This specific order of crits makes for a very fast enemy turn, requires no Itchy Feet, no healing Spawny and Luigi is very close to where the Sentry needs to go, without being in range of the Sentry. This is only possible like this on 1.0

Naturally there is still room for improvement left. For instance, I played super slowly and carefully once I got past 4-6-1. No push crit on 4-6-1 would also save a bit more time.
Other than that I currently have no ideas on how to improve the run. I really put all my game-knowledge to use to make the route as optimized as I could before starting attempts, as I knew this would take a long time.

If you want to do this yourself on 1.0 you might want to watch out for the no HUD glitch. In short if you press Y and then A in too quick succession when activating Extra Health you will start the battle with no HUD. This itself is not a problem, but another side-effect is that you can't skip turns anymore. A way to quickly fix the glitch mid run is to pause unpause with plus.

I spent over two weeks of doing attempts until I finally got this run. I didn't count the total number of attempts, but I tracked the first 500.

Here are the first 500 attempts:
o = nothing
i = bounce
u = push
b = burn right
d = burn left
p = push + burn right
q = push + burn left
s = burn to goal
z = push + burn to goal
? = didn't pay attention / messed up execution
'-' = break in between attempts

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Using this data one can estimate:
Peek-a-Boo crit chance: ~30%
Valkyrie burn chance: ~26%
Chance for Spawny to reach goal once burned: ~1.35%

Furthermore I collected some data on 4-6-2 that suggest:
Hopper burn chance: ~42%
Chance to meet 4 conditions: ~84%

This gives
Chance of successful 4-6-1 = 60% * 26% * 1.35% = 0.21%
Chance of optimal 4-6-1 = 60% * 70% * 26% * 1.35% = 0.15%

Chance of successful 4-6-2 = 84% * 60% = 50.4%
Chance of optimal 4-6-2 = (1 - 50% * 42%)^2 * 30% * 70% * 70% * 60% = 5.5%

Chance of successful 4-6 = 0.106%
Chance of optimal 4-6 = 0.008%

Feel free to use the date for some analysis of your own!

I hope you enjoy the run :)