I love to see a young scientist in training taking a whack at a thorny philosophical/psychological problem. Kudos. That said I take a harder line then you and do not believe there can be such a thing as a reasonable mistrust of the scientific process. Saying you do not trust science is tantamount to saying you do not trust facts. Not trusting facts is not reasonable. Of course you may reply, but Dan I did not say science, I said the scientific process. By scientific process I take you to mean the scientific method. If that is not the case then many of my objections will miss the mark however if it is then your obligation is to describe and define the scientific process as you see it and make me see how it is different from the scientific method. Moreover, you must be able to do that without appealing to the human element (i.e. you cannot just say scientists are people therefore it can be distrusted because people can be distrusted). You began your piece by saying that people can have a reasonable mistrust of the scientific process, humans/scientists are not required for science to be as it is or for the scientific method/process to function the way it does or work as well as it does. They are that way because the universe is that way and that would not change if we were not here to observe it. Of course you are free to disagree with that as well but I would then argue you have abandoned a core scientific principle of the existence of an objective, independent reality, that exists no matter how many or how few (or none) conscious beings may exist to observe it. Quantum phenomenon do not change this in any way for reasons others have described in great detail elsewhere.
As I do not believe you can or would do any of the things I described in the preceding section I will use the terms scientific process and scientific method interchangeably for the remainder of my reply. Your next approach may be to say, OK Dan I will grant that the scientific process and method are one and the same thing but the scientific method is not the same thing as science. I agree that not trusting science cannot be reasonable but that is not the same as not trusting the scientific method. If you decide to go down that road you will need to define science and the scientific method for me in exact, specific, detail in such a way that the difference between the two is made clear. You might start by arguing that one is a method and one is simply a thing. I once had some sympathy for this view. On its face it seems obvious and incontrovertible. Let’s ask some questions about this view and see where it leads. If the scientific method is not science what is it? It is the method of doing science. What is a method? It is a way or a system or a set of rules that define how something is to be done. Therefore the scientific method would be the way or the set of rules for how science is to be done. Those rules have been established and proven to work over and over and over again, in fact they have never failed when applied to a problem that is within the purview of science. Why do they never fail? Because they are scientific in nature. It is the nature of the scientific method to not fail when applied to a problem in science, it cannot fail or it is not the method. In other words the scientific method is scientific in nature. What is scientific in nature is science. Therefore the scientific method is science. Therefore since it is not reasonable to mistrust science it is not reasonable to mistrust the scientific method/process.
Finally you may take one last shot and try and object that, in certain circumstances, and/or for some people, mistrust of science could be a reasonable position. I am running out of steam so I do not intend to go into great detail on this one but ultimately the only way this argument can (appear) to succeed is through an appeal to belief. It might be reasonable for people to believe that science is not trustworthy however it can never be reasonable to distrust science. For example someone who is insane might believe he is no longer bound by science and scientific principles, and that for instance he can fly. Because he is insane that belief is perfectly reasonable to him, though of course it is not to the vast majority of other persons. Additionally, the truly ignorant or woefully misguided may believe that science is untrustworthy. To them and others like them this belief is reasonable. To the objective universe, and in our current existing reality, it is not.