Meaning of "little heart" vs. "less heart"? and why

Solution 1:

Less is the comparative form of little. The original sentence:

He has little conscience and less heart = He has little conscience and [even] less [than a little] heart.

The rewrite doesn't make sense because you aren't comparing less conscience to anything:

He has less conscience and little heart

You could say, however:

He has less conscience than heart.

Solution 2:

Could I solve you problem? That all depends on what your problem is.

Little conscience and less heart means he does not have much conscience, and he has even less courage (heart) than he has conscience.

Your alternative is difficult to understand, because you mention less conscience, but you do not tell us what you compare it to.

So, in short, no, your alternative is not good.

He has little consciences, and (even) less heart.
He has little heart, and (even) less conscience.

These both are fine, although they do not mean the same! Twist the order of the parts of the sentence, and you will be less readily understood :)