I believe Jesus was fully God and fully man, yet the Bible states he clearly was tempted. It leads me to wonder, did the man, Jesus, always know he was God, or was this something that he learned, or gradually developed. I can't imagine God being tempted by anyone or anything, therefore, there must have been some point in Jesus' development where he came to this realization, or am I missing something?
Since the Bible doesn’t clearly answer your question anything I can say is speculation based on what the scriptures do say. I think we can make reasonable guesses about how much he knew and how much he could do.
At least as early as age twelve Jesus appears to have known his origin and his mission. When his parents found him in the temple discussing the scriptures with the rabbis he said, “Did you not know I must be about my father’s business?” (Luke 2:49) It is also hard to imagine that his parents did not tell him the stories about his birth and early life in Bethlehem.
On the other hand, there is Paul’s statement in Philippians 2:5-11. “Let this mind be in you, which was also in Christ Jesus: Who, being in the form of God, thought it not robbery to be equal with God: But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men: And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross. Wherefore God also hath highly exalted him, and given him a name which is above every name: That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth; And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.” This seems to indicate that there was at least a period of time when he divested himself of the glory and power of God and became, if not fully man, at least essentially a mere man. I recently heard one preacher speculate that he had no power to perform miracles until after the Holy Spirit came upon him at his baptism. We know of no miracles before then.
The temptation began after his baptism, though, and according to Luke 4:13 may have continued throughout his life. Certainly when he was tempted in the wilderness he knew who he was and had the spirit and power of God.
As you say, it is hard to imagine God being tempted. “Let no man say when he is tempted, I am tempted of God: for God cannot be tempted with evil, neither tempteth he any man.” (James 1:13) Yet “we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin.” (Hebrews 4:15) The argument made by the writer of Hebrews is that the sacrifice of Jesus would be worthless if he had not been able to sin or be tempted in the same way we are. Perhaps this is where the “wholly God and wholly man” theory comes into play. Although he was God, he was also man. While he could not be tempted as God he could be tempted as a man. I don’t fully understand it. I just thank God that it happened.