I would vomit, I would struggle, it might be the worst experience EVER but without a time cap I would eventually find the way to eat the whole pigeon. If you said 20,000 dollars I probably wouldn't do it because is not only disgusting but also very difficult. However, I think 10,000,000 dollars are worth the try. There are people who do much more horrible/hard things for much less money after all.
PS: That makes me think... are you even allowed to vomit it? You did swallow it after all, I think not vomiting it (after it reached your stomach) is pretty much impossible...