There are two aspects I disagree with here.

The first is formal relevancy. These questions only make sense if we already assume we have a theory of everything. We know we don't. From a formal perspective the totality of the formal theories behind modern science are inconsistent. So results like Godel's incompleteness theorems don't apply formally.

Okay, but what if we do get a theory of everything? What happens if a string theorist comes out and says they've solved it? Well let's examine the Game of Life, where we do have its theory of everything (the game rules) and it happens to be undecidable. In particular, given an initial configuration, a pixel X might be eventually shaded in if and only if ZFC is inconsistent.

Let's assume the game is playing in a screen. So is an investigation into whether pixel X on the screen will be shaded scientific?

This brings me to my second point. Popper would probably say yes. This is a classic example of the problem of induction that is falsifiable. Kuhn would probably say he needs more information about how the study is going to be carried out and the underlying metatheory of the researchers. Lakatos probably would say something similar to Kuhn in this case.

Yet I have a hard time believing any of these people would be hung up on the decidability part of the problem. As an empirical phenomenon whether pixel X on the screen will be shaded seems little different from whether a bowling ball on a plank will start spontaneously levitating. It probably won't, but oh boy will it be interesting if it does! And crucially in both cases I think, given the right setup, most everyone could agree that staring at the pixel or staring at the bowling ball for a long long time constitutes a scientific experiment, albeit a boring one.

Perhaps the objection might be raised that these are different. We don't have a theory of everything behind the bowling ball, but we do have a theory of everything behind that pixel.

Well if your theory is fundamental enough to completely obviate the need for empirical experimentation, I don't think many people would agree the subsequent logical work would qualify as science.

To put it another way decidability and incompleteness are not very helpful sociological explanations of science.

There's nothing stopping someone from saying that the definition of science is intrinsically tied to decidability. After all people can make whatever definitions they want.

But it doesn't seem useful to me, because it doesn't explain what it is that people who call themselves scientists applaud and what they frown on. That is it's not a very useful descriptivist framework. That is whether a question is decidable or not does not seem to be the criteria of whether a question is reasonable or not.

Indeed let's not talk about hypotheticals.

What do scientists who are familiar with the incompleteness theorems have to say? Well Hawking talks about it: http://www.hawking.org.uk/godel-and-the-end-of-physics.html but doesn't use it to split the scientific from the unscientific. In fact he seems to derive joy from the incompleteness theorems.

Indeed I have a hard time finding people who view logical completeness or decidability as the differentiating factor between pseudoscience and science as ways of thinking and acting, rather than as potential limiting results on what the theory of everything would look like.