How did the Buddha know he wouldn't be reborn?

Yes, special knowledge is required to utilize the conditioned as raw material:

“An arahant monk,
one who is done,
effluent-free, bearing his last body:

Would he say, ‘I speak’?
Would he say, ‘They speak to me’?”

“An arahant monk,
one who is done,
effluent-free, bearing his last body:

He would say, ‘I speak’;
would say, ‘They speak to me.’

Skillful,
knowing harmonious gnosis
with regard to the world,

he uses expressions
just as expressions.”

—Samyutta Nikaya 1.25

“This fivefold method of mastering perception is called in Pali:
ariya iddhi, a term which may be rendered as noble power, noble
success or noble magic; or, alternatively, as the power, success or
magic of the noble ones (ariya). In its perfection, this arduous
practice can be ascribed only to Arahats as several suttas and
commentaries indicate. But, as our text shows at the beginning,
the Buddha recommended this training to the monks in general,
including those in whom the three unwholesome roots were still
active. It is eradication of these roots which is said to be the
motivation for taking up this practice.”

—Nyanaponika

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This is correct. Craving ceases when ignorance is suppressed. Through insight meditation.

Ok, but if conditioned phenomena ceases when craving ceases, then what is the conditioned experience of an arahat dependent upon? And how can one check it by oneself in the experience of insight?

Alternative to phenomena ceases, probably better to say seeing impermanence….then the truth is seen and then it follows then thereafter craving ceases, and not any moments before.

Samsara obviously.
Nibbana with residue Iti 44:

And what is the Unbinding property with fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. His five sense faculties still remain and, owing to their being intact, he is cognizant of the agreeable & the disagreeable, and is sensitive to pleasure & pain. His ending of passion, aversion, & delusion is termed the Unbinding property with fuel remaining.

This part of the OP…

…appears to be approaching things from a different perspective than these suttas:

SN12.15:
“Kaccāna, this world mostly relies on the dual notions of existence and non-existence. But when you truly see the origin of the world with right understanding, you won’t have the notion of non-existence regarding the world. And when you truly see the cessation of the world with right understanding, you won’t have the notion of existence regarding the world."

(world a.k.a. self)

DN15:
It wouldn’t be appropriate to say that a mendicant whose mind is freed like this holds the following views:
‘A Realized One exists after death’;
‘A Realized One doesn’t exist after death’;
‘A Realized One both exists and doesn’t exist after death’;
A Realized One neither exists nor doesn’t exist after death’.

MN38:
“Sir, [consciousness] is the speaker and feeler who experiences the results of good and bad deeds in all the different realms.”

“Silly man, who on earth have you ever known me to teach in that way?

SN12.20:
When a noble disciple has clearly seen with right wisdom … it’s quite impossible for them … to turn forward to the future, thinking: ‘Will I exist in the future? Will I not exist in the future?

So, I’m unsure if it’s accurate to say that the Buddha “knew that he wouldn’t get reborn”. He didn’t phrase it in that way afaik.

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Craving < Feeling < Contact < Six Sense Fields < Name and Form < Consciousness < Volitional Formations < Ignorance

As per dependent origination.

The destruction of ignorance is the knowledge that all rebirth has come to an end. That no future body will be taken up. So, how?

When one stops craving for sense pleasure he stops experiencing feeling. When feeling is gone, he loses contact with sense media. He does this by withdrawing from sense media.

He tears asunder the attachment and craving for sense media. That is the fuel which allows name and form to burn.

But, having detached himself from contact and feeling with regards to sense pleasures, he eliminates the fuel that will determine his rebirth in a new body after death.

But, there are rebirths that happen without the body. Rebirth in the dimension of “Consciousness” for example. This is also relinquished. And then volitional formations end, and with that, supreme knowledge of the goal is attained.

He does not have fuel for rebirth in a body. He does not have fuel for rebirth in formless realms. He cannot even generate the fuel, because he no longer craves feeling and contact. That is how he knows. It is not merely the feeling or the contact which has stopped. It is the “attachment” to those things.

Without the condition of craving, the whole chain of DO unravels.

But what becomes of the Arahant. Well. Does he exist after death? Does he not exist after death? Does he both exist and not exist after death? Does he neither exist nor not exist after death?

None of these questions can be answered. They don’t apply to the Arahant.