What you suppose is your “internal” morality compass is an “internalised” one.
I wouldn’t kill my grandma because I love her, not because it’s “wrong”. I won’t kill anyone, I guess, because I don’t like seeing living beings suffer. Not because it’s “wrong”.
Morality is always an internalised “system”. It can’t be “natural”, it’s always ideological.
But that doesn’t mean that being materialistic in analysis of our existence as humans would make you do “evil” things. If you try to analyse us as a species scientifically, we realise that we literally evolved to cooperate and be nice to each other. Our chemistry makes that necessary. We hate being alone and seeing those around us suffer, because those things produce “feel bad” chemicals. We love helping each other because that produces “feel good” chemicals. On average of course (as you mention psychopaths do exist).
In fact, a purely material analysis of us would show that greed, individualism, destroying the planet, killing all animals on it, making large portions of our species to suffer in poverty etc. are counterproductive. Those things all make us individually feel worse and have worse lives. We would have the best lives if everyone around us had access to all amazing developments of the past centuries freely, if the animals and ecosystems of the world were protected, if the people around us cared about us and lived with us, not despite us. And none of that is moral, or based on morality. Just science and materialism.