Your question presumes several false things about science, which will make the answer to the first question fairly simple.
1a) You presume that science laws are absolute, and always unbroken. This is an invalid assumption. All science "laws" are only regularities, and even the most "fundamental" of them spontaneously break at times. This is made explicit in this paper on symmetry breaking in physics, and the spontaneous breaking of all the fundamental "laws" of physics: https://www.pnas.org/doi/10.1073/pnas.93.25.14256#:~:text=Symmetry%20principles%20play%20an%20important%20role%20with%20respect,structure%20and%20coherence%20to%20the%20set%20of%20events.
1b) You further presume that science is complete, and there are no known violations of any of the "laws" of science. This is blatantly not the case, all active science fields are incomplete, and there are known violations of nearly every regularity that are called laws in science. Exploring those violations, and trying to figure out if they can be explained by a modified set of regularities, is what science DOES. All active science is therefore a process of studying and providing explanations for "the supernatural" per your definition.
1c) You presume that science "laws" are coherent, and they therefore do not violate each other. This is manifestly false. We do not have a valid reduction structure to fundamental physics, some fraction of physics is not reduced, and only about half of chemistry, and only aspects of biochemistry in biology. Outside of these fields, complete reduction has been unsuccessful in science. PARTIAL reduction remains highly useful, as SOME causes can be found to be explainable at a reduced level, such as chemical poisoning of organisms. But the existence and stability of organisms, for example -- is only understandable at a higher level of biological system theory. And things like species, ecosystems, ecological niches, norms, societal institutions, etc. -- none of these seem to be reducible even in principle. They just don't rely on their substrate in any way, they exist due to logic phenomena and structures at a higher level. The goal of being able to reduce all of science to physics has been abandoned as unachievable in principle by the vast majority of philosophers of science. See the SEP article on reductionism in science, section 5. Science has adopted pluralism -- the subjects studied by other sciences besides physics are real, and their phenomena are causal.
One of the consequences of pluralism is radical incoherence. IF one accepts multiple uncoupled sources of authority, they WILL end up with conflicting predictions in their derived consequences. Hence, pluralist science CANNOT EVER in principle create a network of "this is consistent with science, and that is not".
Your second question is how to provide support for miracles, and in particular for design inferences.
2a) In one respect this question is trivial. There are two sub fields of science that focus on design inference, both anthropology, and SETI. Both reference "naturalism" by trying to distinguish designed objects from natural ones. One need not assume that the "natural" is constrained by unbreakable "laws" to do this, regularities work just fine for both fields. Key to both fields is to presume an agent as goals, and the powers/capabilities of the agent are critical to identify what one will then look for as evidence of artifacts left from accomplishing those goals.
One can extend this process to theistic design, and this was usefully done for design vs evolution in the TalkOrigins forum debates. TalkOrigins established that evolution fits our evidence set from species far better than design does, but the result for discussions on abiogenesis, and the origin of the universe were less definitive.
2b) You need to understand how hypotheses and theories are evaluated in science. You tend to think in terms of proofs -- a formal logic process. But science operates of a much looser and more pragmatic methodology. Philosophers of science have characerized how this process works, and the best characterization is by Imre Lakatos, who identified Research Programmes as how science operates. See https://www.liquisearch.com/imre_lakatos/research_programmes#:~:text=Lakatos%27%20model%20of%20the%20research%20programme%20aims%20to,abandoned%20or%20altered%20without%20abandoning%20the%20programme%20altogether. Research programmes are a set of working assumptions, which have core and secondary elements, and the secondary elements may be adjusted as evidence is accumulated in the field. Contradictions to the programme are not disqualifying, so long as there is continuing work to try to resolve them, AND there is useful continuing progress in the field using the programme. But when the field is generating more and more unresolved discrepancies, and rationalizations to try to explain them away become the dominant activity in a programme, it has become regressive, and a new programme needs to be developed in the field.
Accordingly, any design "miracle" programme has to show utility in solving problems, and effort to resolve falsifications. If a design programme does that, then it becomes a credible working model, and it will satisfy what you are asking for.
2c) As an aside warning, there are some trends of thinking in contemporary science that are far less useful than Lakatos' model, and these tend to be adopt Bayesian thinking. Bayes developed consistent and coherent math on how to show the change in validity of a theory in the circumstance of accumulating evidence. Bayes math is fully coherent and supported, unlike the standard statistics approaches used in P tests, T tests, etc.
The standard statistics embedded some judgements into how to do tests, and what thresholds are useful, which are USUALLY useful, but occasionally give whacky results. It is a pragmatically created model, that is not internally coherent.
Bayes model ALSO involves judgment calls, but he makes them explicit, and at user discretion. These involve the determination of the "prior" probability" and the weighting of new evidence. The assumption behind standard statistics is that users can't be trusted to make these judgements, so a standard should be pre-set for everyone, even if the standard can't be formally justified. Bayes statistics allow one to openly specify one's assumptions. The trouble with Bayesian statistics, is that the ANSWER is DRAMATICALLY dependent on these opinion-based judgement calls. If one just sets the confidence in one's "prior" very high, then one can honestly evaluate massive amounts of contrary evidence, and still show that it does not change the conclusion, and that the evidence fails to reach statistical significance.
Bayesian statistics in practice end up being used to rationalize not changing one's views in the face of contrary evidence. All Biblical Creationists, for example, are basically Bayesians who consider the prior evidence for Biblical accuracy to be so strong, that no contrary evidence from any science can shake their conclusion. And the recent tendency in science to adopt Bayesianism gives these sorts of dogmatisms a shield or cover to hide behind.
It is not only Creationists who have adopted this sort of anti-evidence rationalization based on Bayesian thinking. The organized skeptic movement has also adopted it. Susan Blackmore, in her autobiography, notes that she will dismiss ayn and every test done for psi phenomenon, because she considers the "prior" of physicalism to be so likely as to be unquestionable -- while the logical possibility of experimental error or researcher cheating is always be possible, for any experiment, hence her beliefs are irrefutable by any evidence, in principle. Similarly, the creator of the Science Based Medicine web site and movement, Dr Steven Novella, will dismiss any positive results supporting any clinical success for alternative and complementary medicine, because there is always the possibility of experimental error or confounding effects in medical studies. He even more radically advocates for a BAN on funding any medical experiments on CAM, on the assumption that all positive results are experimental anomalies, and lead to confusing both doctors and patients about the uselessness of CAM. The SBM web site explicitly advocates for use of Bayesian statistics, with a prior is so high that it excludes CAM for every showing any statistical validity. Organized skeptics, similarly to creationists, can become immune to evidence and rationalize this dogmatism with Bayesian thinking.
