MWI really has nothing to do with the Fine-tuned Universe paradox. That paradox has to do with the fundamental physical constants being just right to allow for

MWI really has nothing to do with the Fine-tuned Universe paradox. That paradox has to do with the fundamental physical constants being just right to allow for life. But the many worlds in MWI aren't worlds in which the fundamental constants are different, but rather they're worlds in which quantum probabilities collapsed in different ways. In fact, of al the infinitude of worlds in MWI, they all must share the same physical constants, because, for every quantum decision, only physically possible results are represented by parallel worlds, and there's no quantum mechanism that can change the fundamental physical constants altering reality. Given that, you can easily reject MWI while still accepting the parallel universes explanation for the Fine-tuned Universe paradox (namely, that there's an infinity of universes with different physical constants, which solves the paradox quite neatly because only in universes with constants "tuned" just right for life can there arise life capable of thinking of the Fine-tuned Universe paradox, which turns the paradox into a straightforward example of the Anthropic principle). Or you could reject the parallel universes explanation for Fine-tuned Universe while still believing in MWI for quantum effects.