What do you do when you're about to touch a stove that might burn you? You naturally look at your hand at the point of contact as you're about to touch the stove.
What do you do when you're about to touch a dog that you're afraid might bite you? Here, you might also start looking at the point of contact as you're about to touch the dog because you're afraid that the dog will suddenly bite your hand. In other words, you broadcast that fear to the dog and that fear of being bitten can quickly engender a self-fulfilling prophecy. If you're afraid of the dog, the dog will be afraid of you. Fear begets fear.
In my opinion, this is what you're doing right now. Nothing has happened yet. But you're so afraid that something might happen, that you might be the one to trigger the backlash yourself.
This is currently a non-story. Let it be that. This person is not your CEO, nor your celebrity spokesperson, nor your talk-show-host. You're not paying this person millions of dollars for representing your organization. You don't have a morality/political clause in your contracts. Your organization is not some huge corporation that can afford to redo work that has already been done.
Furthermore, paying him and redoing his work anyway (as Magish suggested) would still be a huge insult to him and may create the backlash you're worried about. At the most basic level, this guy is still a human being, he does the kind of work he does for the money, yes, and many other selfish motivations, most likely, but just like any of us, he also does it for wanting to feel useful. Rob him of that and you only demonstrate a profound lack of empathy towards him.
Again, do not focus on what might happen. Do not focus on your fear. Release the group project. Do it properly as you would any other project. Do not hold back on publicizing the project on your end. Give the person the credit he deserves (I assume he did a good job for his part).
Do not try to hide or obfuscate his identity (as someone else suggested). Do not even mention that he try to use a different version of his name. If he wants to have some of his work to appear under one name and some other work appear under a slightly different name, that's his prerogative and that's his call. But it is not something that you should even suggest to him.
As to ignoring him, you shouldn't ignore the obvious either. You can tell him you ran across the other project he created. You can start a conversation about politics and about the underlying intent of your group project. Considering that your work content is highly political and agenda-focused. I think it would only be natural to ask such questions. Just don't use the group project as an excuse to scold him for his views. And don't imply that you'll all be boycotted over this, or that your own image has been tarnished by your association with him. The real truth is that you don't know.
Do not brace for impact. Don't try to do damage control preemptively. Deal with any complaint as they come over time (if they come at all).