This is why you'd use gravity braking to do the work for you... You just have to change direction, not slow yourself down, as long as your path intersects the sun you're done... we have sent spacecraft to the sun... it's not beyond our technical capabilities. Basically, it's like flying into a banked curve... as long as you're pointed at the Sun when you exit the curve you're going to hit the sun with out needing to…
The difficulty does not come from the mass of the Sun. It comes from Earth's velocity around it. The waste starts out with that velocity as well. Basically the waste is already in a stable orbit around the sun, and getting it out of that is difficult.