I agree that, considered by itself, Liberation of Baghdad might be taken seriously in a GO-USA! way by some jingoistic people. But when seen side-by-side with The President's Dream, that "too cheesy" quality and the general sarcasm become more obvious.
As a more general observation, I would point out that in a domestic US context, some people's criticism of the Iraq War was hopelessly entangled with and colored by their anger over the 2000 election -- in other words, the war had to be evil not simply -- and perhaps not even primarily -- because many innocent Iraqis were killed, but because BUSH AND HIS PLUTOCRATIC RETHUGLIKKKAN NEO-NAZIS ILLEGALLY STOLE THE ELECTION AND GORE SHOULD'VE BEEN PRESIDENT ARGGH BARRGLE GAAAAH!!!!!!!! And analysis of "Bush's War" was sometimes colored by other factors that had nothing at all to do with the pertinent question "Will this invasion of Iraq, in the long run, tend to improve stability in the Middle East?" (The gay columnist/blogger Andrew Sullivan, for instance, was an early supporter of the war, but turned completely against it with rather suspicious suddenness when George Bush endorsed the idea of a Constitutional Amendment to ban same-sex marriage.)
Whether any of this applied to Birk, I can't say, because I don't have a clue about his general political views. But it's important to consider that his paintings supposedly about the Iraq War may have been, on an underlying level, paintings about the Bush Administration. (Of course, this also applies to supporters of the war -- some of whom may have been reluctant to criticize the war policy because they supported Bush's election in 2000 and did not want to ally with Gore supporters, or because they didn't want to be associated with "9/11 Troofers", or whatever. But in short, issues not related to "Does this war make sense as foreign policy?" affected their opinions on the war, too.)