Herod's mental state as it survives in the history by Josephus move than covers whether he would be of the frame of mind to murder these children if he thought it threatened him as the fulfillment of the coming messiah who would rebuild the temple. Josephus records that as his imminent death was approaching, fearing that people would be laughing, he herded everybody into the Hippodrome with orders that they be killed upon his death, so that would assure nobody would be laughing at the time of his death because they would be mourning all those killed. But those in charge decided not to carry this out. But this is more than enough, if true, to demonstrate his desperation.
ANTIQUITIES 17.6.5-6:
5 "...For that he was not unacquainted with the temper of the Jews, that his death would be a thing very desirable, and exceedingly acceptable to them, because during his lifetime they were ready to revolt from him, and to abuse the donations he had dedicated to God that it therefore was their business to resolve to afford him some alleviation of his great sorrows on this occasion; for that if they do not refuse him their consent in what he desires, he shall have a great mourning at his funeral, and such as never had any king before him; for then the whole nation would mourn from their very soul, which otherwise would be done in sport and mockery only. He desired therefore, that as soon as they see he hath given up the ghost, they shall place soldiers round the hippodrome, while they do not know that he is dead; and that they shall not declare his death to the multitude till this is done, but that they shall give orders to have those that are in custody shot with their darts; and that this slaughter of them all will cause that he shall not miss to rejoice on a double account; that as he is dying, they will make him secure that his will shall be executed in what he charges them to do; and that he shall have the honor of a memorable mourning at his funeral. So he deplored his condition, with tears in his eyes, and obtested them by the kindness due from them, as of his kindred, and by the faith they owed to God, and begged of them that they would not hinder him of this honorable mourning at his funeral. So they promised him not to transgress his commands.
6. Now any one may easily discover the temper of this man's mind, which not only took pleasure in doing what he had done formerly against his relations, out of the love of life, but by those commands of his which savored of no humanity; since he took care, when he was departing out of this life, that the whole nation should be put into mourning, and indeed made desolate of their dearest kindred, when he gave order that one out of every family should be slain, although they had done nothing that was unjust, or that was against him, nor were they accused of any other crimes; while it is usual for those who have any regard to virtue to lay aside their hatred at such a time, even with respect to those they justly esteemed their enemies."
The account of killing the children at Bethlehem might have made Herod even more unpopular, so it is possible it was done secretly and thus we have no historical record. But another theory might be that his killing or trying to kill someone from every household is related to his killing these children and we are just seeing a distorted version of that. However, we sill have the historical reference that he was of the state of mind to kill someone of innocence in multiple households.
Jesus was born around September 14-15, 2 BC. When Herod killed the babies, Jesus was already over a year old. Herod died on Shebat 2, 1 AD, 18 days after an eclipse occurred on December 29, 1 BC. Jesus and his family fled down to Egypt after which it was heard that Herod died. So we're only talking about a few months for this to occur, from mid-September through mid-January, though the closer we get to his actual death, the more unstable he became.
It should also be noted that some considered him the coming "messiah" who was to rebuld the temple. This may have been his interest in rebuilding the physical temple at Jerusalem. His followers were called "Herodians." This reflects back on the revision of the chronology. Based on the revised chronology, which is 82 years earlier than the true dating (537 vs 455 BCE), the expectation of the messiah would have occurred 82 years earlier than 29 CE, which is around 54 BCE. Herod was around 20 at this time; he became king over Judea in 37 CE and ruled for 37 years down to 1 AD. His chronology is distorted to expand the rule of his son, Archaelus and so it is claimed that the 37-year rule dates back to 40 and his rule beginning in 37 was only 34 years. Yeah, sure!
At any rate, he certainly would have seen any other potential messiah as a threat to his own claim as messiah and if he killed rivals in his own family whom he was in fear of, I'm sure there would have been no hesitation to kiill a few "innocents" under the age of two in the small town of Bethlehem. So even though we have no extra-Biblical account of this specific event, it certainly fits the context in many ways of something Herod would do to anyone threatening him or his image.