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Suppose we (scientists) say they don't.

8-) That's an argument from authority. It's not always a bad type of argument, but it is subject to error. And it's ultimately a philosophical argument.

How do you enforce your claim of privilege?

The natural sciences rest on metaphysical presuppositions such as the idea that natural laws are uniform and predictable; that the universe is ordered and predictable; that scientists can trust the evidence of their senses; that something cannot both be, and not be, in the same sense and at the same time; that the whole is greater than its parts; that mathematics has objective value and existence; that objective truth exists, etc.

And metaphysical presuppositions, presuppositions regarding the nature of reality, belong to the realm of philosophy, by definition and by nature. Philosophers determine the scope of the natural sciences. Scientists don't determine the scope of philosophy. The medieval maxim is correct: "Theology is the queen of the sciences and philosophy is its handmaid."

You raise some serious issues that I will try to discuss seriously

- that natural laws are uniform and predictable, and that universe is ordered and predictable, are metaphysical presuppositions. I dispute both contentions. That natural systems are uniform and predictable is an empirical observation, and one that is not universal. Quantum mechanical systems in some cases exhibit randomness which we believe to be unavoidable; some dynamical systems exhibit chaos - behavior that is determinisitic but diverges rapidly towards unpredictability. We do seem to have discovered some universal laws, but I submit they are universal only because we haven't yet found counterexamples; not because we make the assumption they are universal.
that scientists can trust the evidence of their senses. There is some justification for calling realism a metaphysical presupposition, but it's not unique to science. The assumption of realism is universal to all human endeavors (except, perhaps, some religions). And I'd make the argument that it's a byproduct of experience as much as a presupposition. When we're infants, we notice that we cry, and mommy comes. Thence we discover that we can affect the world around us, and can do so in ways that have pleasant results.
that something cannot both be, and not be, in the same sense and at the same time. This is an empirical observation, and not a universal one. Before a measurement, a quantum mechanical system is generally in a superposition of states. If you ask if a hydrogen atom is in the 1s state; the answer is yes; if you ask is it not in the 1s state, the answer is also yes. Once you do the measurement, you force the system to 'make up its mind', if you like.
that the whole is greater than its parts Sometimes it is; sometimes it isn't. Empirical observation.
that mathematics has objective value and existenceMathematics has done a pretty thorough job of analyzing its own structure as a mathematical problem. I'm not really competent to discuss that; but I agree that it does appear that mathematics is not a closed, complete system; some small number of presuppositions are necessary. As to whether math exists and has value: again, I'd say that's far more an empirical observation than a presupposition. Some fields of science are highly mathematical; some are still mostly qualitative; the extent to which math. is valuable depends to a great extent on the maturity and the complexity of the field.
that objective truth exists Apparently, yes, there are some universal constants, but their 'objectivity' is a matter of scientific observation. Einstein emphasized that GR, which is all about what are universal, objective measurements and what are based on reference frame, stood or fell by experimental test, not by the beauty or mathematical consistency of its structure. Observationally, there are some entities, like the speed of light, that have an objective value irrespective of reference frame. But the key word is 'observationally'.

To conclude; I agree that science requires a small number of metaphysical presuppositions. Those presuppositions are, however, so limited and so universal beyond science that they don't really support the contention that philosophy (or less yet, theology, which presumes an entity whose existence many scientists don't accept) have some sort of essential underlying structural role in science, or that they determine its scope. What philosophers think of science has seldom had any impact on us at all.