Does physicists' choice of an approximation scheme determine the laws of nature?

Certainly. That is what many "physicists" believe, that is the foundation of much work in "physics". For example in the most widely used approximate calculational method, perturbation theory, certain integrals in intermediate steps have a lower limit of 0 so are infinite. This perturbs "physicists" immensely, so they try to revise nature (which is why they developed string theory which has the great attraction of being known wrong) to eliminate thus "problem". However when all steps of the algorithm are completed the result is finite and correct. The infinities are meaningless and are purely the result of stopping in the middle. If another approximation scheme were used this "problem" would never have arisen. Nevertheless "physicists" feel that this make-believe problem is so important that nature must be changed (for example by changing particles, which do not appear in the theory and have nothing to do with these integrals, to strings). Then there are the absurd beliefs about the vacuum. There is a way of drawing pictures to keep the bookkeeping straight in this particular way of approximating. Some of these pictures can be fancifully interpreted to imply absurdities about the vacuum. But there is a difference between bookkeeping and physics. Unfortunately "physicists" do not understand this. They will claim it is just their extreme incompetence. Reading the "physics" literature shows so many examples of the difficulties "physicists" have in dealing with the world of reality (or maybe it is just fraud).