First, it’s worth noting (and this is at least implicit in @MauroALLEGRANZA’s comment, that for Leibniz the PSR is something like an axiom. More precisely, our ability to understand “the universe” depends on the fact that there is a specific reason “why everything is the way it is and not otherwise” (this is Leibniz’s most common formulation of the PSR). Without this, there is both nothing to be explained and no possible explanation.
Second, the parallel that you attempt to draw between scientific laws and the PSR seems to fail. It’s fair to say that for any universe in which any set of physical laws holds true, we can always ask why these laws and not others hold. But note that invoking this question depends on holding the PSR to be true; i.e., the question is only sensible if we already believe that there must be an explanation or reason for why this is the case. Similarly, asking “why there is a universe where the PSR is valid” already assumes that the PSR is valid, that is that there is a reason sufficient to explain why the PSR holds and why it’s opposite does not hold.
To truly try to “get behind” the PSR, we really have to ask (and this is Heidegger’s question), “why is it that we assume there is an explanation at all”? But this necessarily depends on suspending the idea that we could explain why we remain convinced that there must be an explanation. In short, we don’t hit a short-coming of the PSR per se, but of the very structure of any “why” question at all (i.e., why do we think it’s possible to pose any “why” question at all?). (Note that even these questions seem to implicitly depend on the PSR). So ultimately we end up stuck at the level of asking whether it’s sensible for us to ask “why” or “why not”. After all, if there are no sufficient reasons, then there are no reasons why any set of affairs are the case, nor any reason why any set of affairs is not the case. But explaining how this can be the case, without invoking some form of the PSR seems to be impossible (and this, as I read him, is Heidegger’s point). Or, in short, the PSR is necessary for there to be any sort of explanation whatsoever.
Where we hit the paradox I think your aiming at is by asking why we assume that any sort of explanation is possible. But what is at stake here isn’t something that we can explain without already assuming that “explanations” are possible.
I don’t know if any of that helps, but there are some random thoughts.
PS there are a good number of arguments against causality, but these aren’t necessarily arguments against the PSR per se. To argue against the PSR requires conceiving of “reality” (or whatever else) as somehow immune to explanation, which holds for none of Sextus Empiricus, Nagarjuna, nor Hume (Wittgenstein is a much more complicated case)