Upvote:4
As you say, it sounds very unlikely.
Mahmud had previously proclaimed independence in May 1919 after earlier accepting a British protectorate. He was captured by the British and exiled a month later. He was recalled in September 1922 to help stabilize unrest in the area, and proclaimed independence in November with himself as King. In December, the British and Iraqi government issued a joint proclamation recognizing "the rights of the Kurds living within the boundaries of Iraq to set up a Kurdish Government within these boundaries". Mahmud did not respond to this and organized attacks on the British with Turkish support. He was driven out of Sulaimaniya in March 1923 and finally surrendered in the summer 1927.
This chronology leaves just one month between Mahmud's declaration and the British/Iraqi declaration. Given that the British policy under Percy Cox to incorporate south Kurdistan into Iraq predates this, it is difficult to see how any official recognition would have been given. Officially, Mahmud's administration was only suspended in February 1923, which may have been interpreted/misrepresented by some sources as recognition. However, given that the recognition claim does not seem to be mentioned in any non pro-Kurdish edited source that I could find, I would guess that it is a political fabrication.
Some sources: The Middle East in the Twentieth Century by Martin Sicker and A Quest in the Middle East: Gertrude Bell and the Making of Modern Iraq by Liora Lukitz.