No Joe, I very much doubt that it's Irish. The character "The Dodger" has got London written all over him. The tour circuit for these early musichall artists included Scotland and Ireland that's all.
I wouldn't discount an earlier American song though. As I said Sam Cowell was raised there. It would have to put it before 1840. Sam Cowell was born in 1819 and George Coppin in 1819. Coppin toured in America for the first time in 1844, the same time as Cowell's decision (he was in England by then) to concentrate on musichall instead of theatre or opera.
There was an overlap of a few years when Coppin and Cowell could just possibly have both been in the same area of the UK. 1840-1843. After that Coppin went to Australia and Cowell became a great early musichall star in England. Cowell toured America just at the start of the Civil War and was very popular.
I'm still seaching for the "We're All Cheating" words. I'll post them as soon as I find them. Cheers Joy