Learn how to dip your mooring line eye through his, or vice versa. That way either one of you can cast off without disturbing the other's lines.
I'll try to describe it to you; if you saw it you'd see how in an instant.
Imagine his mooring line eye on the cleat. It has an opening in the eye. Take your line's eye and pass it down through the inside of his eye, towards the deck of the pier. Then you just bring it up and back over his eye and onto the cleat. Now either of you can take a strain on the lines and still be able to cast off. Your eye might bind a bit coming free but nothing that a tug with the winch wouldn't free easily. And neither of you will have to so much as touch each other's lines. When you see ship's tied up in a line along a wharf this is how it's done, where their head lines and stern lines share a bollard or cleat.
“Scientists are people who build the Brooklyn Bridge and then buy it.”
Wm. F. Buckley, Jr.