Hi Anya. I've never heard of rooting Mango from hardwood cuttings. You might be able to root newly lignified greenwood (current season's growth but hardened off) using a mist system or some similar home spun method.
When I lived in S. Fl, the only propagation method I heard of for Mangoes was grafting or to grow them from seed. You could store your Mango cuttings until you're able to graft onto a Mango rootstock. I wouldn't suggest storing them in the fridge since it's a tropical species.
If you do store them, here would be my suggestion. Clean them. Dry them thouroughly. Wrap them snuggly in plastic wrap (saran wrap). Zip them up in a ziploc bag and store them in a cool, dry place. They should store for months in your home like this.
I could be wrong about getting your Mangoes to root like fig cuttings. So hopefully someone will set me straight if that's the case.