Take note of what species of birch the local nurseries are selling. Some birches are very disease prone in certain climates, the ones the nurseries sell, will tell you which species of birch do well in your climate. The North American paper birch, native mainly to the northeastern tier of USA states and most of Canada is prone to get attacked by bronze birch borer insects when planted in regions that are warmer than its native range. I think the European weeping birch, which doesn't weep much, is resistant to the borer, but I am not sure, so check what your local nurseries are selling. This will give you the name to look for. Then on line you can find smaller size stock, if you don't want to risk chopping a 6 foot nursery tree.
I understand about aspen. I believe there are very few in bonsai pots because they are difficult to work with. The roots will abandon a trunk you are trying to get to ramify branches in favor of little suckers.
I have no personal experience with chopping back a birch. So I really can't say. My gut feeling is they will back bud, when cut in winter or when cut after first flush of growth has hardened off, right around the summer solstice. (before the middle of July)