Bishop Benn's solicitors have said that the BBC relayed "factual inaccuracies".
A report on BBC South East Today, broadcast on Tuesday, revealed some of the contents of letters between Lambeth Palace and the director of children's services at East Sussex County Council, Matt Dunkley, and the chairwoman of the Local Safeguarding Board for East Sussex, Cathie Pattison - letters obtained under the Freedom of Information Act.
A letter from Mr Dunkley and Ms Pattison to Dr Williams, dated 16 May, said: "We have no confidence in the judgement and conduct of Bishop Benn, in relation to dealing with safeguarding issues, and believe it is appalling that the Church seems reluctant to take decisive and immediate action. Bishop Benn faces serious questions of, at best, his competence to ensure children are protected, and at worst, his gross negligence in the face of serious allegations against staff he was responsible for. His position as bishop is plainly untenable until these matters are fully investigated."
The letter also said that "insufficient attention" was being paid "to the ongoing and immediate safeguarding of children in Sussex".
A response from Lambeth Palace, dated 7 June, said: "Any process of suspension involves us jumping through several hoops and we are by no means certain that the evidence for such a step will be sufficient. I would add that the Church is not like other organisations in terms of employment arrangements."
A statement from Bishop Benn's solicitors said: "Legal obligations prevent the Bishop of Lewes from using confidential information within his possession for correcting those factual inaccuracies mentioned in the BBC report."
Bishop Benn is understood to be nearing retirement, although an official announcement has not yet been made in the diocese.
The letters mentioned in the BBC report were written before the publication of the Archbishop's commissaries' interim report, which said that "dysfunctionality" in Chichester diocese was preventing adequate child safeguarding.
A Chichester diocesan spokesman said on Wednesday: "East Sussex County Council have properly raised matters of concern with the Archbishop of Canterbury's office, whose responsibility it is to comment at this stage. We would refer to the interim report of the Archiepiscopal Visitors, which contains substantial recommendations regarding the Clergy Discipline Measure and the issue of neutral suspension."
A joint statement by Mr Dunkley and Ms Pattison, issued after the BBC broadcast, said: "The letters reflect our long-standing concerns and frustrations about the way in which Lambeth Palace was investigating the handling of safeguarding matters within the Chichester diocese. The letters speak for themselves, and while the language used in them reflects the fact that they were not intended for publication, it does accurately reflect our strength of feeling."
Mr Dunkley and Ms Pattison "agree with the recommendations" of the Archbishop's commissaries' interim report, particularly those that "attempt to bring the Church's safeguarding procedures into line with all other organisations that work with children."
Lambeth Palace declined to comment.