If you're going to have the crate in the run, I'd only have her out during the day (at night, put her directly on the roost, or move the crate into a shed or garage) and I'd see if you can offer some draft protection by perhaps loosely covering a couple of the sides (any facing into prevailing winds) with cardboard, an old towel, etc. Don't want to cover it up completely so she can still see the other birds and still has good ventilation. Being out in the cold by itself isn't an issue but being unable to take shelter from winds/sleet could be.