I would like to answer this question from the perspective of a high school epistemology student, which hopefully can offer an insight though this might be a wishful thinking on my part.

Humanity tend to believe that there is external world out there, be it that we perceive it directly or that we only perceive the representations of it, in the sense that sense stimuli are processed in our brain and we obtain the picture, the representation of the reality that supposedly exist independently of our existence. This is often argued on the view that our perceptions are way too uniform most if not all the time for it to be only a conjuration of our minds, though this intuition on uniformity itself is questionable. In a way, we find it more appealing to argue that external reality exists instead of our minds establishing the reality that we seem to experience.

The main issue that I detect here is that there is no logical necessity for both models to be true; there is no principle that could necessitate external reality nor 'internal' reality created by our minds. Assuming external world exists, we cannot access it, i.e. noumenal in nature. This noumenal nature of external reality already implies that there are infinitely many ways in which we can think of the reality-in-itself (reality's ding-an-sich) in which we would still have no contradiction in our empirical observations and experiences. Assuming the world is created by our minds, i.e. the world in our head, there is no contradiction either; there is nothing that forbids uniformity of sensory inputs and outputs even if all minds conjure their own realities. The coincidence of having identical representations among each person cannot be used for an argument against "the world in our head", for coincidence itself is not logical but merely phenomenal (just a phenomena that requires no logic to operate(?)).

Once we allow this demarcation to dissolve, we end up in a situation where we cannot decide whether external world exists or not. Physics can only deal with observations of the world and not the world in itself; regardless of how this reality-in-itself actually manifests itself, Physics will not change for the noumena is not within the domain of physical study of the world. In a way, it is the method of inquiry that forbids us to make sense of the 'external reality', or rather, there exists no method of inquiry in which this matter can be inquired. Asking about noumenal world would be meaningless, and forcing an answer out of it would allow all sorts of probable interpretations and solutions to be devised and still make sense to us humans.

Why can't we solve BIV problem? It is because the context we are in, i.e. we are the brain in vat, robs us the method of inquiry in which we could know that we are brain in vat. How this robbery occurs is very simple; one just have to put the subject into a non-third party perspective, i.e. we are not one who observes the brain, and we will then have no way of knowing we are brain in vat. Infinite regression seems logical to us and asking for its resolution seems absurd; but precisely that because it is within the realm of logic and we cannot avoid logic that it does make sense; logic is the tool of inquiry that tells us about the unsolvability of infinite regression and existence of external reality and BIV problem. I would not be so bold as to claim that in principle there exists a way to solve them if we can extricate ourselves from our logic that bindsus in this plight, but this claim itself is based on logic and thus its conclusions must be logical by nature.

Hence, I think that it is precisely the fact that we are in the non-third party perspective of the world that we cannnot access the 'external world', and this in turn results in arbitrary possibilities of conceiving what it is. It works like mathematical axiomatic systems; the axiom is "there is external world independent of our experience", then we begin building knowledge systems which conforms to this axiom, and Physics works precisely by taking this axiom as a necessity - that if this 'axiom of external existence' is true, physics will make sense. For the independence of the external world from us would suggest that objectivity is plausible concept and uniformity of nature can be understood without invoking coincidence, which in turn implies that this axiom suggests to us that what we termed as Logic makes perfect sense and comes naturally to us as the necessary tool to study the world around us.