I don't see how it would be different from using any other MIT-licensed code in your project. The Apache License doesn't seem any less restrictive than MIT at first glance?
Edit: I assumed the entire codebase was multi-licensed, it looks like you're saying that different parts have different licenses, which does sound like a headache.
Edit: I assumed the entire codebase was multi-licensed, it looks like you're saying that different parts have different licenses, which does sound like a headache.