That phrase in verse 6 is part of 12 whole verses that are omitted in many modern translations. You do not ask about that, but in one Interlinear of John's gospel I found this comment about the Greek text of those 12 verses, which gives a clue about that phrase. It reads:
"[Manuscript] Aleph BSy8 omit verses 53 to chapter 8, verse 11, which
read (with some variations in the various Greek texts and versions) as
follows:..." (K.I.T. p.460, 1969 edition)
The phrase in question in ch.8 vs.6 is not in the Greek text copied in smaller print as a footnote in the K.I.T. It looks as if the manuscript used did not have it, but also that other manuscripts do have it. Interestingly, another Interlinear which uses the Nestle Greek Text, literally translated by Alfred Marshall, and including the text of the Authorised Version, does not have those 12 verses at all. However, given that the Authorised Version, and other translations do, while others do not, the simple answer is that some manuscripts have that phrase, while others do not. Therefore it all depends on which manuscripts have been decided upon.
There is no doctrinal reason for omitting, "as though he heard them not". Observation of the scene described would clearly show Jesus' would-be accusers asking him that question, with his response being silence as he bent down and began to write on the dusty ground with his finger. So, as verse 7 states, they asked him again. At that point, he straightened up and replied, "Let the one of you that is sinless be the first to throw a stone at her."
He had heard the first time but chose not to answer immediately. They had asked him whether he agreed with the Mosaic law, that being caught in the act of adultery required that death penalty. His astute response of initial silence, writing something in the dust with his finger, caused a pause. The stones were no doubt all around, ready to be picked up in an instant, but some time elapsed with nothing happening. The question was repeated, and Jesus straightened up, speaking to show that he had heard the question perfectly well, for he answered so as to get his questioners examining themselves, and not the woman, nor himself.
In that way, it's clear that Jesus appeared not to have heard their question the first time when he'd heard it both then, and the second time. He deliberately responded in such a way that no stones started flying. The real intrigue (for me) is, "What did he write in the dust?" That writing was just as deliberate as his initial silence, and no doubt the older men noticed it, and its significance, because they melted away first.
You ask if there was deliberate deletion of that phrase, "out of fear that Jesus' behavior of feigning ignorance of the question loudly put to him, could perhaps send a wrong message to the critics of his divinity". No, because whether that phrase is included (as per some manuscripts) or omitted (as per other manuscripts), the divinity of Christ is not affected one way or the other. Jesus deliberately chose silence as his initial response, for a reason. The highly-charged atmosphere was then calmly changed by Jesus' responses, for he took control of the situation and completely changed the outcome. If anything, that all contributes to upholding the status of Christ, so that one wonders why some modern translations have chosen to completely leave out all 12 verses, never mind that one little phrase, without even mentioning the event.