Lodging options are few (maybe none) at Buttermilk and their base has very little to offer compared to Snowmass Village. Also, Snowmass has the new Kids' Treehouse. I would stay ski-in/out at Snowmass and just do the kids' lessons there. You definitely don't want to stay in Aspen and have to schlep to Buttermilk each day (and Ajax is not where you want the kids in lessons vs. Snowmass or Buttermilk).
As for lodging, find the Snowmass Village map, do a search at a site like this http://www.ski.com/
(I saw some really good deals on this one) or this http://stayaspensnowmass.com/
and pick your price point based on location and rating of the unit. There are still more shops and restaurants in the mall level (original) village vs. the new base village, so staying near there is convenient and you can just ski down to the Treehouse (or ride the SkyCab). I have found that these sites also have the best deals on lift tickets if you book with your lodging.
We are going back to the Snowmass Mountain Chalet this spring break, but it only has hotel rooms, not the 2BR condo you are looking for.