The short answer is that you generally don't fire executives because of the issues it creates. You usually force them to step down, or you just don't renew their contracts.
See when you get fired at your job, that's just sort of it. HR comes in, you sign some stuff, clean your desk, gone. No one knows anything. You have no contract, you can be fired for any reason - and as long as you can't prove it was a wrong reason, you can't do anything.
When executives get fired, there is usually a contract. They can't just fire them willy-nilly, there has to be a reason. And when you hit that upper crust, usually the firing needs to be addressed in a shareholder's meeting, making it very public. So if you fire an executive for a non-ironclad reason, they have a very solid basis for a lawsuit.
If Iger does want to go into politics, he's not going to want to have to explain firing a top woman at Disney, especially one as connected and vindictive as KKK. Its less that it would derail his aspirations, and more that its an obstacle he doesn't want to have to deal with if it can be avoided.