First of all, I genuinely appreciate your engagement with my arguments. So thanks for that. Now, to the business at hand.
You say: "Or are you arguing that science can be performed without the need for naturalism?"
The answer is yes. It is yes because I include reason as necessary to the truth seeking enterprise about the entire universe. Whether the subject is the physical world of sub-atomic particles in energy fields or language and information, mathematics, economics, etc..., reason is the Sovereign of truth. I use a capital "S" for a reason and it's explained in an earlier post. Reason is the ultimate authority of what is and is not true. Think not? Then argue with me about it and you are forced to reason. Facts are not "subject" to reason. They just are. But the explanation of those facts, aaah, now that's another story. Any explanation of anything that contradicts itself cannot possibly be true. Let me pick on the poster child for scientific irrationality, Richard Dawkins, who says that there is NO PURPOSE IN THE UNIVERSE on the one hand and then writes books about how the struggle to surive is at the bottom of the evolutionary process. Well, which is it? How is "struggling to survive" not purposeful? How is his writing a book not purposeful? His metaphysical claim is false or his empirical claim is. Actually, in this case, they both are false but that's another story. How anyone can take that guy seriously is beyond me. Way beyond.
If science is about making inferences from data (reasoning from effect to cause, essentially) then reason is part of the natural world. But they say it isn't. I don't get that.
The first question is ontological. What exists? If we get that wrong, everything else is wrong. But we can get that right through the exercise of pure reason. With apologies (not really) to Kant. But science rejects reason in that capacity yet relies on it for "scientific" explanations. Seems odd to me.