Muse Cut was the 4th flourish I ever learned. I used the tutorial by Henrik Forberg
I am assuming you are doing the same.
You can hold the deck two ways. One way leads you further into the flourish. IIRC, the other way feels more comfortable, but makes working the second packet next to impossible.
Anyway, you say you're having trouble with the packet and the deck passing each other. I tried out Mise Cut a little bit and I believe I may have an answer for you.
Part of it involves 'lowering' the deck as much as you can with your pinky and ring finger. The lower you can get, the easier it is for the packet to pass. You also need to have a light grip on the packet edges so if the packet makes contact with the deck, the packet adjusts accordingly. But the BIG part of the flourish relies on the grip you have with the pinky and ring fingers. If you can nail that, the packet-flippery won't be NEARLY an issue. The flips are the easiest and most fun part of the flourish.
Get that grip! And enjoy the flips!
I hope this helps in some capacity, let me know if you need additional help.