I'd go the full crampon / ice axe route for safety if it were me. Better to have and not need than the other way around.

For maximum weight savings get a pair of light weight aluminum crampons like the Petzl Leopard FL since you are focusing on snow travel and a lightweight Axe like the Petzl Ride. Combined the two would weigh in at 21 oz or 600g. I took a look at all the offerings from Grivel, Camp, BD, and Petzl for axes and crampons and those two look like the lightest combo.