Tl;dr: Physicists would care about which set theories to choose iff there is an observable difference between the choices. That's less likely than you think, because "observable outcomes" are always finite numbers, but nevertheless, somebody should look into that (indeed they do).
Almost all mathematical physicists would agree that this is the only reason to care. For the sake of rigour, they implicitly choose the most common set theory ZF(C) due to time constraints.
Let's first give an answer to the following question:
Do we need to care about the foundation of set theory in physics?
Detour to the scientific method
Science is in many ways a social and personal endeavour (see my detour below). This is crucial, because it influences the way people think, but in this section I want to quickly sketch a few more "objective" ideas - in that they are more or less universally accepted ideas: The scientific method.
The "objective" point of science accoring to that method is predicting outcomes. Those can be past outcomes, but it should also work for future outcomes. This is why science is "useful". Because of the complexity of life, this has to be broken down to small units - experiments. The nature of experiments is all but simple and it is certainly more than "observation", because our senses are not very good (think about optical illusions). In order to remedy this, we invent measures and measuring machines so that, in the end, the outcomes of experiment is usually a bunch of numbers on a computer screen. That means all outputs are always computable.
Most physicist view mathematics as a tool. They marvel at the effectiveness in physics, but all they want to do is to approximate physical theories. In fact, if you don't work at Theories of Everything, you are well aware that you only work with effective theories. An effective theory is by construction only an approximation. Many mathematical physicists work on effective theories themselves and they agree.
At that point, I would argue, the question of whether we need to consider the foundations of set theories depends on the world view. Here are the most common alternatives:
1st idea: Since everything we ever measure is computable numbers and the number of excitations of quantum fields is likewise finite and computable (due to the fact that the universe is deemed finite and everything we think we know about it), a "true" theory of everything would actually only use computable numbers.
If we agree on natural numbers (and even the set theories you cite normally do), then this means the question of foundations is irrelevant. But obviously, we use functional analysis and the like. Why that? Well, because approximating a finite but huge system by an infinite system is much more convenient. Things become much simpler and we avoid a lot of technical hassle. But at this point, it doesn't really matter which set theory we choose. The Banach-Tarski paradox in ZFC? It's just an artifact of our approximation and has no bearing on the real world. Why should we therefore only use ZF or another system?
2nd idea: A true description of the world needs the continuum, however, anything we can ever accomplish is an approximation to that true description or even many different approximations that for some reason cannot be welded together. The foundation of set theory may matter here - but to a very limited degree: It matters if and only if two theories would lead to observably different outcomes. Since, once again, all experimental data we ever get consists in computable numbers, this observable outcome must be in computable numbers. I guess most mathematical physicists would see this as unlikely and thus focus on other matters at hand.
3rd idea: Some people believe that the universe is a mathematical structure in need of the continuum and this structure can be found. At that point you need to care about set theory. But once again, you need some way to discern the "correct" set theory foundation by experiments. This should be possible if you believe that the strucute can be found.
In any case, a mathematical physicist will only care for the foundation of set theory, if there are observable differences between two choices. Since all the data we can ever have consists in computable numbers, this seems like a long shot and thus most people don't really care. Some do - you might like the article "Set theory and physics" (paywalled, but there are versions without paywall that I don't link because they go directly to the pdf) and other papers by Svozil. Note that he tries to find observable quantities that differ depending on the outcome.
Does that answer your question? I'm inclined to believe that no, since everything I said so far does not only hold true for mathematical physicists, but for every physicist. The refined question is probably:
Why don't mathematical physicists care about the foundation of set theory to be rigorous?
If you only want to be rigorous to be sure to have a consistent system, you could pick any consistent formulation of set theory. Of course, consistency is nice, because we like to believe that our world is consistent, hence inconsistent theories would be wrong. Hence to achieve that goal, you can choose whatever theory you want - and it's most convenient to do what everybody else does, namely ZFC.
However, that's not the only reason why mathematical physicists believe that rigour is necessary. It might also help to better understand the theory. Examples are given in the answers to this question: The Role of Rigor
Once again, for all of that, it's enough to use the most convenient formulation of set theory as long as you choose any.
The last objection I can think of is:
But mathematical physicists rarely state which set theory they work with. Shouldn't they do that?
My answer to that would be that 99% of all mathematicians don't do this, so why should mathematical physicists? Why don't mathematicians do it? Let me give a short final detour:
Rigour in mathematics:
Pure mathematics is rarely rigorous. Reading an advanced paper in, say, Functional Analysis, you'll have to fill in a lot of gaps and the foundations are rarely specified. The closer you come to the foundations of mathematics, the more rigorous the papers become (by nature of the endeavour), but even then, many parts of the proofs are regularly left out.
Mathematics as personal endeavour
The problem is that mathematics is a human and therefore also a personal endevour after all and for most people, it's a question of gaining knowledge. However, every person - unless forced otherwise - has some level at which he/she considers things "obvious". If you want to "know" anything, you'll have to make assumptions at some point, otherwise you go down the rabbit hole and end up nowhere. Decartes famously ended up at "I think, therefore I am", but it's not even clear that this deduction is valid.
So you have to start somewhere and you could start at set theory. However, most people will not naturally start there, but consider things like "natural numbers" obvious (you could call this "naive set theory"). They won't really care about whether this is actually consistent or not until they hit a contradiction. Other mathematicians are happy to choose just any formulation and work at higher levels, mostly the one they were taught at. As long as they gain "intuition" about what's going on, they see it as gaining "knowledge" about the field of research. This is also why they leave out certain arguments in proofs - just thinking that they know how it would be done is enough to them (and they are wrong quite often!). This is as I understand it one of the main points of Bill Thurston's famous essay "On proof and progress in mathematics". There is also the question of time constraints: Not everybody can start at the foundations, otherwise no one can build the higher theory.
Incidentally, the personal nature of mathematics is also why many mathematicians feel uncomfortable about proofs done by computers. Using computers to check proofs is fine, but to find proofs? What does the mathematician learn?
Mathematics as social endeavour
Second, mathematics, like everything else, is a social endevour. Both mathematicians and physicists often like to deny this, but research goals, the way research is done and the like are heavily influence by social norms, I will likely do research similar to how my teachers do research. Of course, differences in style will develop, but they will usually be incremental. The biggest changes in how research is done is usually done by a very small minority of extremely idiosyncratic thinkers like Hardy.
Otherwise, most changes in the level of rigour are due to the fact that something affects the outcome of research: The nineteenth century saw a more rigorous analysis of convergence, since for instance it is easy to construct a class of functions $f_n$ which converges pointwise to another function $f$, yet the arclength of the functions $f_n$ does not converge. Since intuitively we think this shouldn't be the case something is "off". Similarly, paradoxes have let to axiomatic set theories in the 20th century.
I am sometimes dismayed by this unfortunate lack of foundation, which is why I endorse algorithmic proof checkers and the like.