Having been in similar situations that you're currently in ... first off, I would say the most important thing to do is to keep your composure. From the tone of your post, it seems like there's a degree of urgency to this, and you really want to be able to give them what they want and still have a good mix. And that's good.
But keep in mind that most musicians are basically turds.

Or at least the ones that work with me. I'm a musician, and I'll freely admit I was once a turd. Now that we have that out of the way, remember that you were hired to mix, I'm assuming, and not master. So what would be so wrong with explaining to them that you may have reached the limits of your capabilities? That you consider yourself a very competent mixer ... but merely an adequate mastering engineer at best. And their music is SSOOOO GOOOD, that you just don't feel that you can do justice to the greatness that is their music in that regard.
That way, you can just pass the baton to someone else and say : "Here, you deal with it."