All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
This statement was reported by a group of scholars other than those mentioned in the question, among whom is Al-Fasawy in Al-Ma’rifah wat-Taareekh with a chain of narrators whose men are trustworthy according to the conditions set by Al-Bukhari and Muslim.
The reviewer of Siyar A’laam An-Nubalaa’ classified it as authentic from Naafi’ Ibn ‘Umar that Mu’aawiyyah sent 100,000 dirhams to Ibn Umar. When Mu’aawiyyah called the people to pledge allegiance to his son Yazeed ibn Mu’aawiyyah, Ibn ‘Umar said, "Now, I realize what he is after. If I do so (i.e. If I pledge allegiance to his son), then I would be selling my religion at a cheap price."
Its meaning – as evident to anyone who knows the Arabic language – does not entail any criticism of Ibn ‘Umar; rather, it is a virtue for him; as his religion is not so cheap as to compromise it in anticipation of money or in order to achieve a worldly purpose.
The story is that the Commander of the believers Mu’aawiyyah asked Ibn ‘Umar to pledge allegiance to his son Yazeed during his lifetime, so that he will be a caliph for the Muslims succeeding his father after his death, but Ibn Umar refused to do so. When Mu’aawiyyah sent him a gift, and Ibn ‘Umar thought that the purpose of the gift was that he would pledge allegiance to his son Yazeed, he (Ibn ‘Umar) said, "If I do so (i.e. If I pledge allegiance to his son), then I would be selling my religion at a cheap price.” Meaning that if this gift will lead me to pledge allegiance to Yazeed during the lifetime of his father, then my religion is of little value to money. But his religion was not of little value to him as he did not pledge allegiance to Yazeed during the lifetime of his father.
Also, the biography of Ibn ‘Umar is bright and outstanding and far from being accused with such a false and condemned accusation. If one refers to the very books mentioned in the question and to the very contexts mentioned, he will find the opposite of this meaning. Hence, accusing him (Ibn 'Umar) of such things is nothing but following desires, innovation, and misguidance; we seek refuge in Allaah from that.
Allaah knows best.