I'm straying a bit off topic, but while agree on some of the premises, I reach a different conclusion.
I agree with you, and probably more broadly that studying systems which aren't fully isolatable is particularly challenging, but I think there is more than a single simple explanation for this. One we need to get out of the way is that there are things under the psychology umbrella that are science and ones that aren't and what is and isn't science isn't simply topic-based, it's approach-based.
For example, reaction-time is a psychological measurement. Is reaction time scientific? I think that's an ill-posed question. It's, like you mentioned, an epistomological question: specifically an ontological question[1]. In the sense that reaction-time is measurable and largely repeatable with respect to specific stimulus, yes, measuring it and analyzing the results can be a step in the scientific path toward knowledge. I'd find it surprising if most people disagreed with this.
Let's take a more difficult example, is behavioral psychology science? Again, an ill-posed question. Can we ask scientific question of behavioral psychology? Sure; Does the intervention of CBT in anxiety-diagnosed subjects (as opposed to a non-intervention control) result in lower post-treatment hospital admittance? That's a valid scientific question. Does it say anything about how CBT works? No. It relates a treatment to a result, and if we want to be more specific, a treatment at a point in time, for a specific group.
Anticipating the test-retest issue you mentioned above, it's being loose with assumptions and isolation. These things can either be controlled for, or assumed. We should be honest with ourselves that far too many times, they haven't been in practice, but let's not conclude from that that psychology has an essential characteristic of not being scientific. It has a social problem instead.
I think we can all be more transparent about assumptions, conditions, and generalization, but I think that's a benefit, not a disadvantage. It enables us to be more precise in our ideas, concepts, and language and that's a good thing.
Specifically regarding the fixed-point framing. It's an assumption of time-invariance. I'm not exactly an expert on the philosophy of science, but I'd be surprised not to find stronger and weaker versions of that assumption and that's a reasonable thing to be transparent about when communicating validity.
1. As an aside, it's funny that "Is reaction time scientific?" is itself not a scientific question.