It's a switchcraft 3.5 mm plug, .290" opening for cable. I was looking at this ( http://www.mogamicable.com/category/bulk/microphone/stereo/ ) data sheet and it seems to say that the width of the double cable is 4.8mm X 2, which I guess wouldn't fit the switch craft. Maybe your connector is wider? I wouldn't mind using two separate shielded cables if they were thin enough to go through the opening of the mini plug together.
The Doublehelixcables mini is indeed wider. Stripping off the insulation prior to the plug is an option to get things to fit, but it won't be as clean looking. Take a look at the Mogami install cables. They tend to have less filler and thinner but tougher jackets. I'm fairly certain you can find something that will work. You could also drill out the back of the Switchcraft to make it larger.