Arguments/premises to disprove God's omniscience
My refutation of God's omniscience consists of showing the impossibility of omniscience itself. That addresses the core of your question and more specifically your 2nd premise, for an entity (be it God, a person, an oracle, etc.) does not have something that does not exist ... that would be a non-sequitur.
Omniscience is just a property ascribed to God. No one knows whether
it is true or not.
As I will explain, the 2nd statement is false.
If I ask an entity "What is the very next number after ²√2 (the square root of 2)?", the entity will be unable to come up with the correct answer. Indeed, for any number that the entity purports to be the answer, I can rule it out by producing other numbers which are even closer to ²√2 (and, thus, likelier to be the very next number) than the entity's answer. One such number is the average:
(²√2 + entity's_answer)/2.
Our trivial ability to compare any two numbers x, y (with x!=y) in the set of reals R suggests that there must be a very next number. The difficulty lies in finding it, but the mere impossibility of computing it cannot negate its existence. That is, the very next number exists; it just cannot be computed or identified. Therefore, that number cannot be known.
The existence of at least one item (here, a number) which exists but is unknowable implies the impossibility of omniscience. This leads to the conclusion that omniscience, the property ascribed to God, is untrue.
An omniscient being would need more than a second to record and store
data about the entire universe. So there will be gaps in the recording
The very notion of omniscience implies determinism and negates randomness. Omniscience implies that an omniscient being would necessarily possess proficiency in deterministic techniques with which he could compute the state of a system at any given point in time.
Let f=f(t) express that deterministic algorithm or function. Determinism renders any recording gaps inconsequential. That is because information about the state of the system at that any particular instant can be recovered from the deterministic formulation f anyway.
But the conclusion about Statement #2 unveils the actual problem; namely, that one cannot know what happened immediately after that specific instant. This means that the pseudo-omniscient entity can figure out what happened at t=²√2 by computing f(²√2), but not what happened immediately after t=²√2 because the entity does not even know what input t to use in f(t).
To be omniscient, a being would require to capture and store all the
data in the universe. This would need a space larger than the universe
Since omniscience is impossible (see above), the amount, availability, and purpose of storage of the totality of data are inconsequential and/or pointless, accordingly.