Okay, I figured it out myself and I'm a moron.
The solution is to apply the "Arnold Properties" modifier to the object you would like to manipulate. In this case I wanted the object to exist to the objects around it but create no alpha.
After applying the "Arnold Properties" modifier to the object, I checked "Visibility" and unchecked "Camera (Primary Rays)" under the General Properties tab. But I kept all the other properties in this section checked. Works great.