Unless I'm mistaken, a high intake of any protein will exacerbate gout, regardless of whether it's animal or plant-derived. I stand to be corrected on this point though.
My concern about intaking large amounts of mass-produced red meat are the residual amounts of anti-biotics and hormones and other crap that are fed to these animals. If this is a concern for you, go organic.
Also, you might consider making sure your meat is the leanest available, to limit your intake of saturated fat.
Have you consindered making your own ground beef? You buy a hunk of chuck or brisket, something that has some fat on it but not a lot. Then half-freeze it and then chop it up in a food processor. It's dead-easy and I bet you'd be able to save a shitload of money and have fresher, better-tasting meat. Also, with this method you're eating meat from only one individual animal, whereas a package of ground beef you buy in supermarkets is a combination of meat from many different individuals, which increases the risk that you will be poinsoned by e-coli or other contaminants.