While not what you want to do (cut through the framing), it probably won't result in imminent wall collapse, either. The styrofoam in the wall should maintain enough rigidity to prevent movement. Since you're basically stuck with that hole where you cut it, it might be possible to attach reinforcement around the hole, connecting the sections of the frame member and still run your dryer vent through it. If possible, tie the cut member (top & bottom) to nearby framing for additional stability.
This would be my guess on what to do. Heartland may have a more professional idea (or not).