It's too late, but would it be possible to move your interview? I was recently asked to come in for an interview.
They said: "Can you come in on Wednesday for an interview?"
I was in the middle of a really big deadline and responded: "I have a deadline on Thursday, but am available on Friday of next week." (not even the next day, but next week! How cheeky!)
They wrote back: "We'll see you on Friday."
You obviously have something that they're interested in. They might be flexible in your interview time if you ask.