“Survival of the fittest” doesn’t work with much of anything INCLUDING the facts of evolution.
Making this the basis of social relationships is of course the biggest disaster of all justifying abuses in capitalism as well as the genocidal behaviors of both communism and national socialism.
But it doesn’t work all that well with evolution either, for cooperation is one of the most successful survival strategies. That is why replacing “natural selection” with “survival of the fittest” doesn’t work, because the selection for survival is not just something that works on individuals separately. Furthermore, it is a misconception that evolution is driven by natural selection. It is driven by variation and natural selection is just a filter. Thus far from inhibiting evolution, the community protecting the weaker members of society accelerates evolution by allowing a greater variation of individuals and thus enabling individuals to take specialized roles in the community. This is necessary for all the technological developments of the community whether it is the community of cells in multi-cellular organisms or human communities.
SO… a more appropriate question is how does natural selection work with the Bible? Because it is correct that this is an important part of evolution. It means death is necessary for the development of life. And this means that in order to be reasonable we should not understand the Bible to mean that physical death is a consequence of sin that will be eradicated. The consequence of sin is spiritual death and our resurrection is to a spiritual body (1 Cor 15). With that it becomes an excellent fit, so that I can say I only believe in Christianity because of evolution. Both require us to accept some harsh realities in which death is required for change, and the creationists and anti-Christians shouting at each other about how the other is too callous becomes rather hilarious.