It should work if you:
1) Select the lamp and the outlined type, group them and copy them as the cut part;
2) Select the orange parts and Path>Compound Path>Make;
3) Also select the lower copy of the cut part and Pathfinder>Minus front (Subtract from shape area in older versions, remember to press Alt/Option if necessary to expand);
4) Repeat 3), only selecting the upper cut part copy and the white rectangle in the background.
Remember to also print the (opaque) white beneath the orange.
It's actually easier than that, Jacob. Since the lamp and letters are "non-printing" color, the artwork can be colored "White" ( knock-out ). But, the Orange and White will have to be modified so that White is actually output as another Spot or process color ( i.e., Magenta or even Black and, then on press, swapped for White ink ). Funny thing about silkscreen is a "Spread" will have to be applied to the Orange. There will probably need to be a "Flash" White applied because of the dark shirt color. But the lamp and lettering will simply knockout of the composite.
It's actually easier than that,
Indeed, and at a superior level, but maybe requiring more explanations, or even enabling errors, by the printer or even in Illy, http://forums.adobe.com/message/2565170