Hallo Kiwis

Here's my non-commercial opinion....
For starters I am not mad keen on Gressoney because the valley is deep, narrow, and does not get much sunlight - not a problem while you're out on the slopes but can be a bit grim in the late afternoon... (JMHO of course). Secondly, skiing in Italy is not very cheap - low cost lodging is hard to come by and eating out can easily set you back a pretty penny.
If I were you I would dedicate a couple of weeks to the Dolomites and a couple to the Alps.
In the Dolomites I would try and stay somewhere near to Corvara. The Superski Dolomiti skipass is not cheap but gives you access to runs throughout the entire Dolomite range - a simply vast area. You MUST have ID (eg passport) and a set of passport photos with you when you buy the pass. From Corvara you can ski to litterally a dozen wonderful places and back in a day. Google it and look at the map

You may be able to find a good lodging solution in a "maso" (mountain hut). Try the websites of the local "pro loco" tourist offices for this.
In the Alps I would try to stay somewhere near Courmayeur. Week-ends are crowded but the views of the Mont Blanc and down the Valle D'Aosta are wonderful even by NZ's Remarkable standards. From Courma you can ski the Mont Blanc itself and make day trips to Chamonix and Argentiere (Les Grands Montets are Something) in France - also to La Thuile which is wide open and windy but with some great runs.
Last but not least... it's just across the border from Cervinia (you can even ski over): Zermatt. The Hotel Bahnhoff is very friendly and comfortable and (relatively) reasonable - but you need to make bookings there errr... yesterday...
Hope this helps