Are there papers or experiments that point to non-fusion sources of solar neutrinos?

For each fusion reaction, we can predict the resulting neutrino energy spectrum. In addition, for each possible source of experimental background (for example, radioactive decay inside the detector itself), we can predict (and independently measure) the shape of the spectrum for that background component. So, when we do an actual solar neutrino energy spectrum measurement (like this one), we get something like this:

The extracted signal components are shown in red, and the background components are all other colors. When we compare the measurement of the extracted neutrino flux of all of these types with the predictions that we can make about solar neutrino fluxes of various types, the predictions agree with the measurements, to within experimental uncertainty. So there really isn't any reason to suspect that another process within the Sun contributes to neutrino fluxes.

For a moment, though, let's suppose that there was some other process. Based on what we have measured, let's see what must be true about this hypothetical process. The main limiter is that there just isn't a whole lot of "room" for a different spectral shape to be added to that plot above. So either:

1. The new process has basically the same spectrum as one of the signal components, and is currently being mixed up with the signal;

2. The new process has basically the same spectrum as one of the background components, and is currently being mixed up with the background; or

3. The new process has a distinct shape, but its contribution is so tiny that it only causes variations on the order of the current experimental uncertainty.

Options 1 and 2 are essentially ruled out by Occam's Razor - it's exceedingly unlikely that a totally unrelated process just so happens to have the same spectrum as something we both know very well and have measured independently in other contexts. So let's talk about option 3.

As we collect more and more precise experimental data, the amount of "room" for a new process to alter the spectrum gets smaller and smaller. This means that the maximum size of the effect gets lower and lower. Of course, this doesn't rule something out, in a strictly technical sense. But, in science, nothing is ruled out, in a strictly technical sense. What it means to "rule something out" in science is different than what it means in other contexts. Scientific conclusions are not logical proofs arising from immutable axioms; instead, they're collections of observations, and new observations can always come along that would alter those conclusions. When we say that something is "ruled out" in science, what we usually mean is that the possibility that it exists depends on such a convoluted series of hypotheses that it's no longer plausible.

This is what happened to the "luminiferous aether" hypotheses in the early 20th century - after Michelson and Morley demonstrated that there was no "aether wind" detectable on Earth, there was of course a significant effort to amend the theory. For example, many postulated that the Earth "dragged" the aether along with it, such that no effect was detectable at its surface, but this would imply that there was a detectable optical distortion arising from the distortion of the aether, which required yet another component to fix, etc. At some point, special relativity became convincing enough in its predictions, given its relative simplicity next to the increasingly-convoluted aether-dragging models, that there was very little doubt that it was the most correct description of nature that we had. In a similar way, it is technically possible to construct a geocentric model of the cosmos that fully explains the motions of all heavenly bodies (this is essentially what many flat-Earthers try to do). It's just that such a model would be so tremendously convoluted, with so many arbitrary moving parts compared to the heliocentric model, that no rational person would be convinced that it was more correct than the heliocentric model.

The same applies here - you can postulate an increasingly miniscule contribution from some other unrelated process, but unless the introduction of that process either a) explains the data better, or b) makes the explanation of other phenomena simpler, it's simply not going to gain much traction. And experiments cost time and money, so it needs to gain at least some traction before somebody puts in the effort to experimentally test it.

Of course, there is the possibility of conducting a different type of experiment to disprove the conclusions of a particular model in a particular parameter space, but how to do this, and whether this has been done, depends heavily on what that other process actually looks like. Without specifying a particular model, there's not much more that can be said other than the above.

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As for detecting "solar-like" neutrinos from terrestrial fusion reactors, there's one huge problem with that idea. Most of the fusion in the Sun is proton-proton (pp) fusion. This kind of fusion has one of the highest temperature thresholds, and is possible in the Sun mainly because the Sun's immense gravity leads to very high plasma densities in the core. We simply don't do pp fusion in terrestrial fusion reactors. The most common fusion "fuel" is a mix of deuterium and tritium - the fusion reaction between those species is orders of magnitude easier to achieve.

It looks like there is another theorized potential source of solar neutrinos: Asymmetric Dark Matter annihilation:

https://arxiv.org/pdf/1606.03087.pdf

Its seems like the new Hyper-K detector coming online soon could potentially detect these, but it's unclear from this paper how abundant they might be.