So, what you seem to be asking is, is the Principle of Sufficient Reason falsifiable. I would say simply, no it isn't.
This is like whether Popper's take on falsifiability, is itself falsifiable, which of course it isn't. And it makes me think more generally of the Problem of Induction: Can we use observations to prove that patterns in observations will continue. Again, logically, we cannot.
I would argue these are about meta-analysis, which is to day are heuristics to think about how we think.
The PSR is not really a claim about the world, but a statement about humans and what explanations mean to us. It frames new information as being capable of being integrated with what is already known, which is to say explained by it or in terms of it, which I'd interpret as a belief in the unity of experience, and implicitly of the cosmos. We see this at work in science, where say the other branches of reality in the Many Worlds Interpretation can be described as not being real on the basis that they cannot be observed or interacted with. It is not the claim that all experience is in the same knowledge or experience domain, but rather that we can generate the means to translate between or interface between domains; for instance using concepts like energy which cross domains as discussed here Is the idea that "Everything is energy" even coherent? Information is another such, and trends in physics are increasingly looking at information as a more fundamental and unifying language than energy, or at a deeper domain from which energy and information transfer constraints like locality are emergent (eg Loop Quantum Gravity).
The universe always gets the last word. And our explanations are limited by our imaginations (see Popper on hypothesis generation). We may very well for all our apparent insights be looking at our cosmos like a dog watching a lecture by Einstein - just incapable of grasping what we are seeing, and what it means. But as humans, it is our defining quality to try to explain.
"Tiger got to hunt, bird got to fly; Man got to sit and wonder 'why,
why, why?' Tiger got to sleep, bird got to land; Man got to tell
himself he understand."
-Kurt Vonnegut, Cat's Cradle
There are types of philosophy which seek to recognise this, and even use this recognition towards generating insights and living well. Discussed here: Philosophers or philosophical traditions that reject symbolic reasoning