History of the phrase "play Mickey the Dunce"
The earliest appearance in print of 'Mickey the Dunce' that I could find came in the form of an appellation, "Micky Dunce", applied by "Silver Heels" to the central character, Michael Cardigan, of Robert William Chambers' historical romance, the 1901 Cardigan. In the first chapter of the work, Michael Cardigan is frequently referred to as a dunce, both by himself and others.
Chambers' Cardigan was widely serialized in magazines and newspapers in the first decades of the 20th century. As his Wikipedia page mentions, by "some estimates, Chambers had one of the most successful literary careers of his period". He continues to enjoy some cachet as one of the originators of the wierd fiction genre.
The character called "Micky Dunce", Michael Cardigan, was presented as Irish. Aside from being the "pet-form of the male forename Michael" (OED), the term 'micky', also 'mickey', at the time Chambers wrote Cardigan, was widespread US slang, usually derogatory, with the meanings "Roman Catholic" and "Irishman" (OED, 'mickey', n.1). At around the same time (OED only has textual evidence from 1909 onward), 'mickey' was also in "chiefly Irish English" (OED) slang use with the meaning "penis".
After the 1901 publication of Chambers' Cardigan, but while it was still being serialized, the exact phrase "Mickey the Dunce" appears in print in the (paywalled) Boston Globe (Boston, Massachusetts) of 22 Nov 1919, p. 5, where it stands in for "the fool", and is collocated with 'played':
… said Mr Dawson. "I played 'Mickey the dunce,' in not insisting…"
Two months later, in the (paywalled) Boston Globe of 27 Jan 1920, p. 1, a follow-up report states that the same Mr Dawson "had played the role of Mickey, the Dunce, all through these oil deals." Similar uses with reference to various people continue in news reporting thereafter.
It would be reasonable to suppose on the basis of the evidence given, and absent earlier evidence, that the term 'Mickey the Dunce' originated with Chambers' use of "Micky Dunce" in Cardigan, as applied by Boston's Mr Dawson to himself, and spread from there.