I wouldn't say that Kevan hates Cersei. He knows that she is incompetent to wield power, but does not actually want any harm to come to her.
Kevan himself does not like playing the game of thrones and would really rather be able to retire from public life completely, but feels that it is his duty to step in a fill the role of Hand and Regent because everyone else who wants those positions would be horrible for the realm.
Kevan was a much better father than was his elder brother Tywin. He genuinely cares about all his children, rather than just wanting to protect his legacy and the family name.
It is important to note that his eldest son is Lancel. Brother Lancel has abandoned the family name to join the Warrior's Sons of the Faith Militant. (Much like the Nightswatch, Kingsguard, or Maesters, joining them requires taking an oath to abandon all titles or inheritance.) Kevan dos not approve of his religious fanaticism, but he still loves his son.
Lancel is of course the chief witness for the prosecution in the case against Cersei. Proving Cersei innocent would mean proving Lancel guilty of perjury. Lancel has already done penance for the crimes of which he confessed, but would face more punishment if his confession was proven false.
Kevan protecting Lancel may require condemning Cersei, even if he has no ill will towards her.
In the books, the High Sparrow was politically savvy enough not to bring any public charges against Cersei for incest with his twin brother leading to the birth of their king. It is sufficient to see her punished for treason and adultery, as there is a strong case that she seduced Lancel and used him to kill her husband King Robert Baratheon. The High Sparrow believes Tommen is a bastard born of incest and should not be on the throne, but wants to keep this secret so as not to support Stannis's claim to the throne. If Stannis still confessed the Faith of the Seven then the High Sparrow would gladly endorse his rule, but he is an apostate too heavily influenced by a Red Priestess who would love to burn all the Septs and force everyone to convert to the service of Rh'llor. If Denarys were already at the gates of Kings Landing then the Faith would likely support her in dethroning Tommen, but they were never too fond of the Targaryens (who were openly incestuous) anyway.
Given that likely only a handful of people in the Stannis army are followers of the Red God, is it that evident that he would force mass conversion if he becomes king? Afaik not even his daughter is a convert (well maybe officially she is, not sure), and Davos doesn't seem to be in any way.