I would think that the only stuff in your water to avoid would be chlorine, chloramine, copper and heavy metals. Stuff like Aquasafe pretty much takes care of that. The softener shouldn't add anything undesirable to the mix.
But yes, about the water from outside - no worries. You should age your water anyway. Let it sit in the bucket for 3 or 4 days. Then when you add it, be sure to do it slowly so that the temperature in the tank doesn't change drastically and the heater can keep up with getting to temp. (I'm assuming you have a heater in the tank; if not, than the water in the bucket should be the same temp as the tank water anyway.)