Frankly, I don't think "shut up and calculate" gets ruled out by PBR, simply because "shut up and calculate" isn't a (single) interpretation, at least in the sense of PBR.
The Fuchs and Peres piece you linked to is a good example of why I say this, but it's far from alone in that ground. The core problem is that "shut up and calculate" (SU&C) does not actually specify whether one should take some form of ontological viewpoint on the subject matter of quantum mechanics and, if so, which one. There are multiple choices there, several of which might be compatible with viewpoints that can be loosely grouped under SU&C; PBR rules out some of them, and it is entirely silent on others.
On the whole, though, I don't really think that the "shut up and calculate" school of thought is really an interpretation at all: quite on the contrary, the spirit it encapsulates is one that explicitly rejects the need for physics to even postulate an ontology for its subject matter. Is the act of actively not-even-trying-to-interpret and interpretation? That's ultimately semantics, but if you do call that an interpretation then I'll just agree to disagree (while quietly seething in a corner about the liberties some people take with language).
On somewhat more concrete grounds, though, if you try to formalize the strict dictum to leave ontology well alone, I would argue that what you get is an operationalist interpretation of quantum mechanics. The best exposition I know of is this talk by Rob Spekkens (starting at around the 51:20 mark, and continued here; Spekkens was aware of PBR at the time, and he gives a more direct response here), but the essence revolves around this mostly-uncontroversial statement about quantum mechanics:
In QM, (i) each preparation procedure $\mathsf P$ is associated with some density matrix $\hat \rho$, (ii) each outcome $m$ of some measurement procedure $\mathsf M$ is associated with a projection operator $\hat \Pi$, such that (iii) the probability of getting outcome $m$ through the measurement procedure $\mathsf M$ after the system has been prepared according to $\mathsf P$ is $P(m|\mathsf P,\mathsf M) = \mathrm{Tr}(\hat \rho\hat\Pi)$.
Most interpretations will then go on and invest the various elements with some form of ontology, but the operationalist approach will stop right at this statement, and hold that there is nothing else to say about the system, including such things as whether there is in fact some system that's produced by $\mathsf P$, and whether it has any properties at all.
If this is what you mean by SU&C, then PBR is completely silent on it. The PBR framework requires an ontological model to work, and the operationalist approach doesn't give it one.
Now, while it's a plausible philosophy to class under that banner, I don't really think that this is what's really meant by SU&C (again, even explicitly refraining from imbuing the mathematical components with any reality is already a good deal more waffling than what I associate with the SU part of SU&C), and indeed e.g. the Fuchs and Peres piece you link to does a great job at playing both sides of that boundary: they start off claiming that there is no need to provide anything beyond "an algorithm for computing probabilities", but then they go on to speak of Cathy's experiment like it actually "exists", and they do that in models that are quite $\psi$-epistemic in ways that are indeed liable to getting ruled out by PBR. However, I don't think that piece is specific enough with its models to tell what it's actually postulating, and by extension it's not specific enough to tell whether PBR impacts its conclusions.
What I think you really wanted to know, though, is not the relationship between PBR and "shut up and calculate", but its relationship to so-called $\psi$-epistemic interpretations: these are realist models that assume the existence of some form of system with some form of properties, which get described by the wavefunction in a strictly 'statistical' way.
If that's what you really wanted to ask, then I personally don't really know ─ but really, when people insist on things like "PBR doesn't rule out any statistical interpretations that are under active consideration", like Steve Byrnes and Ron Maimon (kind of) do here, I really have no idea what kinds of statistical interpretations they do think are worth considering, and I'd quite like to know what they are and how they interface with PBR.
However, if that's what you really wanted to ask, then I would definitely raise a pretty strong objection to the identification of statistical interpretations with the SU&C paradigm ─ which, again, isn't an interpretation.
