You can cut .125" styrene by scoring it once with an x-acto knife and bending it. It will snap right through. The wall cutter won't have any trouble at all.
That said....styrene will last outdoors for weeks. Sunlight breaks it down really quickly. It turns yellow and gets ridiculously brittle. In cold weather? Shatters.
I still use a good bit of it for temporary interior
signs and the .040 material is an excellent substitute of white acrylic in short-term interior backlit
signs. Anything long-term you're going to need a better material.