Should an application for a second postdoc position include a reference to PhD supervisor?
If you're afraid the relationship with your supervisor results in a negative reference for you, I suggest not to include him. However, based on the alternative options you have, you must be sure that they actually know you and are able to provide a reference for you.

Although the director of the institute potentially carries more weight (he's the director, after all), he must know about your work and your work ethic. If you worked closely with another professor in your project, it's more likely he is able to provide a better reference for you, and I would choose the professor over the director in that case.