What should the man of God in 1 Kings 13 have done when faced with the old prophet's lie?

The Mormon perspective on this makes it much easier to understand. True prophets obey the word of God; false prophets do not. In this story are two prophets, one pictured as lying and the other pictured as disobeying God’s instructions. Ellis T. Rasmussen wrote:

“There are some problems in this story of the man of God who came from Judah to warn the king of northern Israel and lost his life in the mission. Some help is available in the Joseph Smith Translation of verse 18, which indicates that the old prophet said, ‘Bring him back … that I may prove him; and he lied not unto him.’ Also there is a change in verse 26, in which the last part reads: ‘… therefore the Lord hath delivered him unto the lion, which hath torn him, and slain him, according to the word of the Lord, which he spake unto me.’ These make the account more understandable. The young prophet should have obeyed God.” (An Introduction to the Old Testament and Its Teachings)

It's a great advantage to have a prophet (speaking both of the story in 1 Kings and of the Mormons in the latter days) to both convey to us the word of God and to help us to understand that word.

I will address Abraham as you have addressed the "man of God"

I believe the issue you are having comes from your interpretation. I don’t have the NKJV myself, but the KJV uses the word “Offer” in Genesis 22, not the word for Sacrifice. This is consistent with the Hebrew at least. I find both versions lack a comprehensive representation of the original Languages. That is my opinion of course on the KJV in general.

“And it came to pass after these things, that God did tempt Abraham, and said unto him, Abraham:and he said, Behold, here I am. And he said, Take now thy son, thine only son Isaac, whom thou lovest, and get thee into the land of Moriah; and offer him there for a burnt offering upon one of the mountains which I will tell thee of.”

If God said “Sacrifice Your Son” which he did not, then we would have a problem. He said ”offer”. This is similar to John 3:16, “For God so loved the world, that he gave his only begotten Son.” The words would have significantly different exegesis if it said “For God so loved the world, that he Sacrificed his only begotten Son” Which it does not say or even elude to as it is his Son speaking about the incarnation and explaining the importance and significance of Baptism to enter into the restored Davidic kingdom of Heaven which is his Church.

Abraham completely and fully fallowed Gods instructions stopping only after the knife was lifted to slay his son, in the mind of Abraham, and so seen by God, The offering was made and Gods command fallowed.

This is one of the richest passages of Scripture foreshadowing the Cross. It is interesting to note that Abraham names the place “Jehovahjireh” (another problem with KJV, as the word means, “The Lord will Provide” and is not stressed in the KJV. The lord did provide a Ram, Abraham did not name the place, “The lorde has Provided” as this foreshadows the future, the lamb of God who takes away the Sins of the world, an offering made to God by God, right on that very same hill in Moriah.

I love the Catholic Church. (opinion again)

There is a key difference between the story of Abraham's "sacrifice of Isaac" in Genesis 22:1-19 and the story of the man of God from Judah in 1 Kings 13.

• Abraham heard directly from the angel of the Lord that he was not to sacrifice his son Isaac after all (see Genesis 22:11-12).
• The man of God in 1 Kings 13 did not hear any countervailing word directly from the Lord or from the angel of the Lord, but from a human being who claimed to be speaking for an angel of the Lord (see 1 Kings 13:18).

The error of the man of God in 1 Kings 13 was that he listened to a human being instead of listening to God.

The man of God knew very clearly what the Lord's instructions to him were. In 1 Kings 13:7-10 he had already refused to stay and eat with the king despite the king's offer of a gift. His words to the king on that occasion were:

Even if you were to give me half your possessions, I would not go with you, nor would I eat bread or drink water here. For I was commanded by the word of the Lord: "You must not eat bread or drink water or return by the way you came." (1 Kings 13:8-9)

But when the old prophet spoke to him and contradicted what the Lord had said to him, he listened to the prophet rather than listening to and obeying the Lord's direct command.

The old prophet who had lied to him also pronounced the reason for the sentence of death by lion:

When the prophet who had brought him back from his journey heard of it, he said, "It is the man of God who defied the word of the Lord. The Lord has given him over to the lion, which has mauled him and killed him, as the word of the Lord had warned him." (1 Kings 13:26, my italics)

Quite simply, the man of God was punished because he had disobeyed God's direct command.

Harsh?

Yes.

But the burden of being a prophet of God was not to be taken lightly, and those were brutal times. God twice tested the man of God as to whether he would obey God's command not to eat bread or drink water while he was on his mission. He passed the first test. He failed the second. Because of that failure, he could no longer be God's prophet, and his life was forfeit.

To answer the question directly: When faced with the old prophet's lie, the man of God in 1 Kings 13 should have ignored the old prophet and obeyed God's direct orders instead.