So I wasn't too terribly surprised to discover the record for a letter Reed had written to Dame Edith in the Manuscript Collection of the Harry Ransom Research Center, at the University of Texas, Austin. And not just a single letter, but a copy of Sitwell's response, and what appears to be a typescript of a 1946 BBC radio program. All three manuscipts are in the Dame Edith Sitwell Collection, 1904-1964. None are dated, and the only references are to box and folder numbers.
Reed's letter appears under the "Index of Correspondents" as Reed, Henry, 1914- --99.2 (Box 99, Folder 2). The section heading states, 'Index entries with no notation (except box and folder numbers) indicate the person listed sent correspondence to Edith Sitwell.' Sitwell's response to Reed appears under the section "Index of Works" as Answer to Henry Reed--1.1. Without some indication of the date, however, it's impossible to divine what the two poets may have corresponded about, although Sitwell did write to John Lehmann in 1944, expressing concern about Reed's article in New Writing (Edith Sitwell: Selected Letters, 1919-1964. Edited by John Lehmann and Derek Parker. New York: Vanguard Press, 1970. 121).
I felt sure that two letters was more than one could hope for, but in doing a
Finally, after trying several possible keyword combinations for London Times radio schedules, I discovered Sitwell (or Texas) had gotten the title slightly wrong: it's "The Poet and His Critic," singular. The serial, a "survey of contemporary verse," ran on the BBC's Third Programme for a brief time from late 1946 to early 1947. Poets included such estimable subjects as C. Day Lewis, W.J. Turner, Dylan Thomas, and Stephen Spender, with critics like L.A.G. Strong, Gerald Bullett, T.W. Earp, and Roy Fuller.
A good, old fashioned Google search put me dead on the money: in the Cleverdon Manuscript Collection at Indiana University's Lilly Library are scripts from Cleverdon's time as a producer at the BBC. Box 18 contains another script for the program in the Sitwell collection, but this one is labeled The Poet and His Critic--The Poet: Edith Sitwell, The Critic: Henry Reed. Nov. 9, 1946.