This question jumbles a few things, which I think can be untangled.
A) It is possible to do science as a nominalist, rather than as a realist. But every scientist I have ever interacted with, is very much of an indirect realist. There may be a few exceptions, but for almost all scientists scientist == scientific realist.
B) Indirect realism presumes there is a reality which we infer by successful models and predictions. Again, every scientist I have interacted with presumes their theories are valid. That is what is proposed when they propose a theory. But when they find exceptions, this causes no massive worldview problems for them. It is the nature of science that it is fault-tolerant -- science does not REQUIRE "exceptionless regularities".
You asked about the "best" science. The best current science holds that all our laws are regularities, based on symmetries, and that all these symmetries spontaneously break. IE those "laws" break too. Here is a description of this thinking, in the form of gauge symmetry theory: https://www.pnas.org/content/93/25/14256
Gauge Symmetry postulates some more fundamental rules that drive the gauge behavior, hence some more fundamental symmetries, and assumes these are "exceptionless", but once more, the theory could tolerate exceptions, without any major problem.
C) Cartwright has a major challenge to accomplish if he thinks that "exceptionless regularities" is logically impossible. It is a burden on claimants of logical impossibility to support their claim, and for the claim to be valid, the majority of reasonable readers would have to agree with the logic argument. I don't think Cartwright has met this bar for demonstrating the impossibility of exceptionless regularities. A discussion of the claim and the validity of his argument would make for a good separate question.
D) Getting to the "universe" vs. "reality". The claim that everything that is "real" or can be "known" about our universe is in the purview of science, is the philosophical position of "scientism". Scientism is a worldview derivable from reductive physicalism. However, the near-consensus of philosophers of science, is that reductionism has failed as a project (See SEP's Scientific Reductionism, section 5) https://plato.stanford.edu/entries/scientific-reduction/.
So SCIENCE itself is "pluralist" rather than reductive. Once one accepts pluralism, then there are multiple fields of knowledge, such as History, mathematics, Logic, etc which are non-scientific, but appear to give us knowledge about the "real universe". Additionally, as science DEPENDS on many of these other fields (math, logic, plus the philosophy of science to justify the scientific claim to epistemological validity) -- the basic premise of scientism is inconsistent with the actuality of science's dependence on outside knowledge.
The majority of philosophers, and the majority of scientists, now identify as "non-reductive physicalists". The meaning of non-reductive physicalism is a bit self-contradictory. If everything is physics (per the name), that IS reductionism! The basic coherence of non-reductive physicalism is, I believe, highly suspect. The meaning, in practice, seems to be a belief that physics is "really real" while other fields, such as math and literary criticism are "real" to perhaps a lesser or derivative degree. There is no "scientific evidence" for this view, it is a purely philosophic position. And the philosophy is self-contradictory, because the epistemology of indirect realism premises that equally pragmatically valid models are equally "real".
But one does not need to be a physicalist of any kind to do science, or accept scientific realism. All one must do, is accept METHODOLOGICAL naturalism. The greatest articulator of how to DO methodological naturalism, Karl Popper, considered physicalism to require reductionism, and reductionism to be falsified, hence physicalism was falsified. Popper's ontological view was there were three worlds -- that of matter, that of experience, and that of ideas. https://tannerlectures.utah.edu/_resources/documents/a-to-z/p/popper80.pdf
So, per Popperian methodological naturalism, math, logic, aesthetics, and literary criticism all reveal aspects of "reality", not just science.
In summary -- science reveals a set of laws that so far all break, but there is no logical necessity for this, it is just a contingent discovery. And science CANNOT encompass all of realty, based on the logical contraction of science relying upon non-science fields. But one can be a scientific realist while accepting both of these truths.