Hard Shell Recommendations (Alpha vs Beta)

I'm looking at getting a good hard shell as that's the last piece of gear I'm really lacking (generic base, REI Fleece, Arc'teryx Atom, thick off-brand puffy), and I'm looking at getting a couple of second hand jackets around 300-400 (Yes, I know to check for fakes/heavy damage). I am in the white mountains so lots of wind but no "Big mountains" and do hiking/ice climbing. I found an Alpha SV and Beta AR for around the same price and yes, I know that the alpha is normally overkill but for the same price as a Beta would it be worth it? It will probably only be used ice climbing and above tree line. Any other jacket suggestions are welcome!