My current understanding of renormalization is that the LSZ formula requires normalized fields. So when you normalize them you get some extra parameters from the regularization procedure you encounter along the way. It's an upgrade on my previous understanding of it as some artificial way of hammering out infinities that arise. However, my current understanding also suggests renormalization is unrelated to perturbation theory, but rather stems from incomplete information, so any nonperturbative method should require renormalization or an analogous mechanism that introduces the same number of parameters. Does that make sense?