If The Holy Bible is inerrant then understanding what is found in Mark (and any other book) must be consistent with everything that is written:

But the Helper, the Holy Spirit, whom the Father will send in My name, He will teach you all things, and bring to your remembrance all things that I said to you. (John 14:26 NKJV)

Applying John means Mark correctly conveyed what Jesus actually said, and while Jesus spoke in Aramaic, the Holy Spirit inspired the words in Greek that accurately record what He said.¹

Of the options listed in Daniel B Wallace’s study (jrdioko’s answer), there is one which permits the criteria of inerrancy and literal accuracy:

• Dominical: Jesus himself made a mistake or was intentionally midrashic (i.e., he embellished the OT story to make his point).

Jesus did not make a mistake. He was intentionally midrashic to make the point He was Lord of the Sabbath and if He is Lord then He made the Sabbath for man.

First, the actual events must be correctly understood in the context Jesus uses. The example of David getting the showbread contains two authorities. One is King David over his men and the other is the high priest over the Bread of the Presence:

So the priest gave him holy bread; for there was no bread there but the showbread which had been taken from before the LORD, in order to put hot bread in its place on the day when it was taken away. (1 Samuel 21:6 NKJV)

There is nothing to suggest Jesus received or ate any of the grain. Rather, He permitted His disciples to do those things. Therefore the comparison Jesus is making is with that of Himself to the high priest who gave the Bread of the Presence to David for his men. The high priest allowed David to take the bread; Jesus allowed His disciples to pick the heads of grain.

However, the priest who gave David the bread was not Abiathar. It was his father Ahimelech:

So David said to Ahimelech the priest, “The king has ordered me on some business, and said to me, ‘Do not let anyone know anything about the business on which I send you, or what I have commanded you.’ And I have directed my young men to such and such a place. Now therefore, what have you on hand? Give me five loaves of bread in my hand, or whatever can be found.” (1 Samuel 21:2-3 NKJV)

Now one of the sons of Ahimelech the son of Ahitub, named Abiathar, escaped and fled after David. (1 Samuel 22:20 NKJV)

When Jesus is seen as using a midrash approach to retell the story He has altered the original event to give Abiathar equal standing with his father, Ahimelech. When this is applied to Jesus in the current situation, the significance is that Jesus has equal standing with the Father.

By stating it was the son (not the father) who gave the bread, Jesus is saying it was the Son (not the Father) who permitted the disciples to pick and eat grain. By calling the son the high priest (not his father), Jesus is saying it is the Son who is the High Priest (not the Father). So it is the Son of Man (not God the Father) who is Lord even of the Sabbath and made the Sabbath for man.

There is no record of the Pharisees response to how Jesus used the Scriptures. However, they could have challenged His defense by pointing out that it was Ahimelech not Abiathar who gave David the bread. Had they done this Jesus could have responded with:

Then Jesus answered and said to them, “Most assuredly, I say to you, the Son can do nothing of Himself, but what He sees the Father do; for whatever He does, the Son also does in like manner.” (John 5:19 NKJV)

Jesus would have concluded His midrash by making the point which is stated plainly in John. Essentially He is claiming equality with God which is a necessity if He is Lord of the Sabbath.

If inerrancy and literal accuracy is found in Mark, then the same standard must be used with Matthew where a similar event is recorded:

At that time Jesus went through the grainfields on the Sabbath. And His disciples were hungry, and began to pluck heads of grain and to eat. And when the Pharisees saw it, they said to Him, “Look, Your disciples are doing what is not lawful to do on the Sabbath!” But He said to them, “Have you not read what David did when he was hungry, he and those who were with him: how he entered the house of God and ate the showbread which was not lawful for him to eat, nor for those who were with him, but only for the priests? Or have you not read in the law that on the Sabbath the priests in the temple profane the Sabbath, and are blameless? Yet I say to you that in this place there is One greater than the temple. But if you had known what this means, ‘I desire mercy and not sacrifice,’ you would not have condemned the guiltless. For the Son of Man is Lord even of the Sabbath.” (12:1-8 NKJV)

The failure to record the name of the priest demands the conclusion Jesus was comparing Himself to Ahimelech. This is at odds with Mark. Therefore, while Matthew’s event has the appearance of being the same as Mark, inerrancy and literal accuracy demands they are seen as two different events. If that is the case then one must come before the other.

If the text is taken literally it includes information to show Matthew is describing the first event and Mark the second:

At that time Jesus went through the grainfields on the Sabbath(s) (σάββασιν). And His disciples were hungry, and began to pluck heads of grain and to eat. And when the Pharisees saw it, they said to Him, “Look, Your disciples are doing what is not lawful to do on the Sabbath! (σαββάτῳ)” (Matthew 12:1-2 NKJV)

Now it happened that He went through the grainfields on the Sabbath(s) (σάββασιν); and as they went His disciples began to pluck the heads of grain. And the Pharisees said to Him, “Look, why do they do what is not lawful on the Sabbath(s)? (σάββασιν)” (Mark 2:23-24 NKJV)

Both Matthew and Mark begin by using the plural Sabbaths: what is described takes place in the grainfields (plural) on the Sabbaths (plural). In Matthew the Pharisees question Jesus allowing His disciples to do what is not lawful on the Sabbath (singular); in Mark the Pharisees point out His disciples are doing what is not lawful on the Sabbaths (plural). Mark can be understood as describing an event that took place on a Sabbath after the one in Matthew.

Jesus first answer was to use the example of David receiving the bread from Ahimelech. He continued and cited the work of the priests on the Sabbath in the Temple and made the point that He is greater than the Temple. On a subsequent Sabbath He repeated the example of David receiving the bread but changed the priest from Ahimelech to Abiathar and omitted any reference to the Temple. Seeing the answers in this sequence enhances the midrash found in Mark.

Recognizing the events took place on two different Sabbaths can bring clarity to Luke who also describes a similar event:

Now it happened on the second Sabbath (σαββάτῳ) after the first that He went through the grainfields. And His disciples plucked the heads of grain and ate them, rubbing them in their hands. And some of the Pharisees said to them, “Why are you doing what is not lawful to do on the Sabbath(s)? (σάββασιν)” (Luke 6:1-2 NKJV)

The perplexing phrase, σαββάτῳ διαπορεύεσθαι can be understood as referring to the event in Matthew

Luke begins by using a different method to describe more than one Sabbath. Instead of making the word plural which indicates Sabbaths in general he describes two consecutive Sabbaths. Then he uses the plural Sabbaths in the question from the Pharisees. Luke omits any reference to the high priest giving the bread; instead David is reporting as taking it:

But Jesus answering them said, “Have you not even read this, what David did when he was hungry, he and those who were with him: how he went into the house of God, took and ate the showbread, and also gave some to those with him, which is not lawful for any but the priests to eat?” And He said to them, “The Son of Man is also Lord of the Sabbath.” (Luke 6:3-5 NKJV)

The omission of the role of the high priest removes the name and the conflict. Thus Luke can be seen as consistent with Mark or even adding a third Sabbath, one in between Matthew and Mark.

It should also be noted that Jesus statement that the bread went to others besides David is also at odds with the event described in 1 Samuel:

Now David came to Nob, to Ahimelech the priest. And Ahimelech was afraid when he met David, and said to him, “Why are you alone, and no one is with you?” (1 Samuel 21:1 NKJV)

Therefore either Jesus is using midrash placing David’s men in the original event in order to make the Pharisees question more relevant to Jesus role as the high priest or there was a second event (not recorded) where David was with his men and took the bread (all of the priests having been killed).