I can't find any info about a US release of OMM by Hoagy. All I can find is the Brunswick recording Stim mentions. That record had "Mr. Music Master" on one side and "Hong Kong Blues" on the other.
Here is a page with an image of that record. Recording date listed is 1942.
In the Podcast page Stim linked it has the following abut "Hong Kong Blues":
Hoagy Carmichael (piano & vocal) with Artie Bernstein (bass) & Spike Jones (drums):
Hong Kong Blues (Carmichael)
Brunswick 03752 recorded Los Angeles, 11 May 1942
Hoagy definitely sings Happy Cat Hit Parade. I Googled that and the only hits I got were references to OMM and pages about cats.
Different story for Lucky Strike Hit Parade. It was actually called Your Hit Parade but was sponsored by Lucky Strike. It ran on the radio from 1935 to 1955.
I wonder now which way it was originally written, my guess is Happy Cat.