Does an enlightened person have access to whatever information he wants?

The Buddha stated that it is possible to be all-knowing, but not all at once.

That means it is possible for the Buddha to know and understand all things if he tried to learn it, but not simultaneously. This means he has the capacity to know all things, but it doesn't mean that he indeed knew all things.

Then the king said to the Buddha, “I have heard, sir, that the ascetic Gotama says this: ‘There is no ascetic or brahmin who will claim to be all-knowing and all-seeing, to know and see everything without exception: that is not possible.’ Do those who say this repeat what the Buddha has said, and not misrepresent him with an untruth? Is their explanation in line with the teaching? Are there any legitimate grounds for rebuke and criticism?”

“Great king, those who say this do not repeat what I have said. They misrepresent me with what is false and untrue.” ....

Then the king said to the Buddha, “Sir, might the Buddha have spoken in reference to one thing, but that person believed it was something else? How then do you recall making this statement?”

“Great king, I recall making this statement: ‘There is no ascetic or brahmin who knows all and sees all simultaneously: that is not possible.’”

“What the Buddha says appears reasonable.
MN 90

But what kind of knowledge does the Buddha definitely possess? According to MN 71 (quoted below), he definitely possessed the three knowledges, and at the same time he denied complete omniscience.

As for knowledge apart from the three knowledges, the Buddha possessed the capacity to know and understand them, but he does not know them all at once.

“Sir, I have heard this: ‘The ascetic Gotama claims to be all-knowing and all-seeing, to know and see everything without exception, thus: “Knowledge and vision are constantly and continually present to me, while walking, standing, sleeping, and waking.”’ I trust that those who say this repeat what the Buddha has said, and do not misrepresent him with an untruth? Is their explanation in line with the teaching? Are there any legitimate grounds for rebuke and criticism?”

“Vaccha, those who say this do not repeat what I have said. They misrepresent me with what is false and untrue.”

“So how should we answer so as to repeat what the Buddha has said, and not misrepresent him with an untruth? How should we explain in line with his teaching, with no legitimate grounds for rebuke and criticism?”

“‘The ascetic Gotama has the three knowledges.’ Answering like this you would repeat what I have said, and not misrepresent me with an untruth. You would explain in line with my teaching, and there would be no legitimate grounds for rebuke and criticism.
MN 71

However, much later tradition exaggerated the claims of the Buddha's omniscience, for e.g. Mil 6.2.8 and Mil 3.6.2. The Milindapanha was authored 400 to 700 years after the Buddha's passing away.

In Mil 6.2.8, Ven. Nagesena was asked why did the Buddha introduce the monastic rules little by little rather than altogether at once, since he was omniscient. He answered that although the Buddha knew all the monastic rules from the beginning, he did not want to scare people with too many rules, so he introduced new rules only when they were needed.

In Mil 3.6.2, Ven. Nagesena was asked the same question, but here he answered that a physician dispenses medicine only after the disease shows symptoms, even if he knew how to treat all diseases.

Mil 6.2.8 and Mil 3.6.2 make the same kind of excuses that was ridiculed in MN 76 (quoted below). So, I feel that it is not good to unnecessarily exaggerate claims about the Buddha, as it will put off intelligent people from learning the Buddha's teachings.

“Sandaka, take a certain teacher who claims to be all-knowing and all-seeing, to know and see everything without exception, thus: ‘Knowledge and vision are constantly and continually present to me, while walking, standing, sleeping, and waking.’ He enters an empty house; he gets no alms-food; a dog bites him; he encounters a wild elephant, a wild horse, and a wild cow; he asks the name and clan of a woman or man; he asks the name and path to a village or town. When asked, ‘Why is this?’ he answers: ‘I had to enter an empty house, that’s why I entered it. I had to get no alms-food, that’s why I got none. I had to get bitten by a dog, that’s why I was bitten. I had to encounter a wild elephant, a wild horse, and a wild cow, that’s why I encountered them. I had to ask the name and clan of a woman or man, that’s why I asked. I had to ask the name and path to a village or town, that’s why I asked.’

A sensible person reflects on this matter in this way: ‘This teacher makes such a claim, but he answers in such a way. This spiritual life is unreliable.’ Realizing this, they leave disappointed.
MN 76