The Bell experiments are widely misinterpreted, but for an understandable reason. What the Bell experiments actually do is establish that if there is a hidden variable, it is irreducibly quantum. It can't be a "classical" hidden variable.

Since the point of hidden variable theory at that point in time was to reframe "quantum weirdness" into "classical sanity", if not outright "classical determinism", and the Bell experiment proved there was no way to do so, the hidden variable theory was then unfit for purpose, and was dropped.

However, it is simply impossible to demonstrate that a particular waveform collapse was not influenced by something beyond science. Consider that if we could prove that, we would therefore also have a proof that reality is not a simulation, but is somehow irreducibly "real" in a non-simulatable way. But we can't prove that; it's too much. (Heck, we can't even really define what that means.)

So the common interpretation that the Bell experiments prove a lack of any external influences beyond randomness whatsoever is not correct, which even a non-quantum physicist can verify by observing that such a claim is simply impossible to ever prove. It is beyond the bounds of what a scientific experiment can prove, even conceivably, so clearly, that scientific experiment did not prove that.

As for how would you ever demonstrate such a thing... well, back down from the loaded term "soul" and let's go with "information source". You prove that an external information source is affecting quantum collapse precisely by demonstrating some set of quantum collapses that demonstrate the existence of information. Of any kind whatsoever, doesn't have to be consciousness; science fiction is full of instruments that extract completely mechanical, physical information from such hypothetical processes. There's nothing really impossible about this, it would just mean our best science and math are incomplete, which is a known thing anyhow (though that doesn't mean that it is unknown in this particular way, but, in general, we know QM is missing something). It's just we've never seen even a hint of such a thing, other than at the fringes of science where the p-hacked phantoms are constantly dancing in front of us but never quite materializing. (See experiments like trying to measure the "world mood" through measuring radioactive decay times; I class this in the p-hacked phantom category.) If I set up a classical double-slit experiment, and instead of the usual interference fringes they started pulsing left and right in a clear Morse code that said something meaningful, that would be that. But we've never seen such a thing, which is why as I said, the default null scientific hypothesis even if we did demonstrate some larger-scale use of "entanglement" in the brain would be that it is still not an information carrier, until demonstrated otherwise.

By contrast, if we lived in a hypothetical world where pretty much every time we set up a double slit experiment it started sending us Morse code messages, and in fact it was only the rare experiment that worked like it does in our universe, we would be somewhat more entitled to assume that whatever our brain is doing might be related to those information sources. But that's so not the world we live in it it is hard to even speculate where we'd go from there. Depends on what the information is. And again, let me emphasize I'm not trying to backdoor consciousness here. Maybe the Morse code is information about the current state of the fusion reactions in the sun or some complicated, but purely physical, galactic resonance of this or that that happens to work out to a Morse code pulse pattern that spells YNK but actually still means nothing much. This is not impossible because it is an intrinsically impossible idea and no universe could possibly run that way. It is instead an idea that does not correspond to what we see.