I would feel very sad, and I would want to do something about it. But I would do so without breaking the NAP. Which means not taking other people's money/labor and letting some third-party spend it as they see fit under the guise of "reducing suffering", even though the goal is noble. I would spend my own money to the degree with which I'm comfortable and how strongly I feel.

This doesn't mean it's a "less moral" system. As a counter-example: Would putting surveillance cameras inside every house to prevent domestic violence be a "more moral" system just because it reduces more suffering? Where do we draw the line for which rights we're okay with violating in order to stop suffering? I just put it at a very non-arbitrary spot, which is the NAP.

That would only work for consequentialists, which Ancaps are not, I believe.

Correct yes - The ends don't justify the means essentially.

https://plato.stanford.edu/entries/consequentialism/

Correct. Ancaps - well, Voluntaryists, if you want to be precise about it - are deontological libertarians.

Consequentialist libertarians seek to maximize individual Liberty. Deontological libertarians seek to minimize immoral actions.