I don’t know if he would have been a fan, per se. But as an Irish Catholic with leftish political leanings, he certainly would have been familiar with the Little Flower’s radio broadcasts.
But I think Rogers would have turned against Coughlin pretty early on – Coughlin started attacking FDR’s New Deal in 1934, by 1936 Coughlin’s anti-communism led him to support Franco in Spain, and between ‘36-38, Coughlin went all-in on his anti-semitic Protocols of the Elders of Zion/Jewish Bankers are Bolsheviks theory.
Given Rogers’ strong anti-fascist tendencies, I think he would have broken with Coughlin in ‘36 if not before.