I used my search box above to find relevant gout facts and discussions. So, I just searched your topic title:
Beer or wine?
Fortunately, my first result was what I’d hoped for: What Is The Best Alcoholic Beverage To Drink With Gout? Also, there are some links to other discussions. But, the main conclusion comes from that article:
The only way to determine if one drink is better than another is to test your uric acid. Stick to one drink for at least two weeks, test your uric acid, then switch to a different drink. Retest after at least 2 weeks, then repeat as often as it takes to get reliable results. If you do this, please share your results in the gout forum, where you can also get advice on establishing test procedures that will produce the best results.
That was also reinforced by a comment in a discussion here: Did you drink wine or beer with your meal?
However, Eric, you are taking allopurinol. In which case it doesn’t matter which one you choose.