Not my area of expertise (if there is one!)...but keep in mind if it is from radon, that is a gas, and I doubt a softener will do much of anything with it. the techniques I've heard use a mechanical means to agitate the water (flow it say in a spray) to promote expelling the radon. Then, you'd need another storage tank and pump to supply the house. If there are actual mineral particles, you might filter them out, but then the filter media would become radioactive! not something you'd want to handle as it concentrates the material.
Hopefully, someone who has successfully dealt with this knows how to resolve your problem.