Very unlikely. The Democrats knew they needed the support of organized labor, but they also knew that to be *too* closely associated with unions, by nominating a union leader for president, would be bad politics in much of the country. Not to mention that the blunt, cigar-smoking Meany was hardly the ideal candidate for a media-conscious age. And after Al Smith, the Democrats figured that they did not want to nominate any Irish-American who fit the traditional stereotype of Irish-Americans the way Meany, like Smith, did. Rather, go for the smooth, Harvard-educated Kennedys.
Besides which, Meany AFAIK never showed any interest in any non-union office.