Why can't you just use a mass driver? Just mine bits of the asteroid and fling them. The biggest problem would be fueling this, and nuclear is probably quite cost effective. (Shout out to KSR's fantastic mars trilogy for this idea.)
Sure, this would be slow. But I think it'd be viable. You could move them into earth's orbit or even slam it into the moon.
A mass driver would run off a capacitor bank most likely - can't you just charge that by a solar array ? Unless you really need to send huge amounts, this should be much less hassle than lugging a nuclear reactor around. And for near-Earth asteroids, you should get the same power per square meter, but better (no atmosphere & clouds to get in the way) at least half of the time (considering the asteroid rotates - and even that could be handled by clever engineering/tethered array).