Not how that works. It's like saying show me the study where they prove god doesn't exist.
I think saying "it's a computer program it can't be sentient" is far too reductive as a general answer, but we know how llms work so we can pretty confidently say that they are really just word predictors. The norns from that game creature someone described in this thread seem closer to protection worthy than llms to me, and the average insect is still infinitely more complex and I don't see many people advocating for insect rights.
Language arises from complex thought, not the other way around. Language on its own can evidently emulate similar results, but since there is nothing behind that that would imply any actual feelings, I don't see a reason to give LLMs any rights.
God does not necessarily mean the abrahamic god. You can insert any non-falsifiable claim there.
Your views seem generally consistent though so at some point it's just a matter of subjective morality (and I do agree with your views on insects and crustaceans in agriculture actually, I'd just put llms far below either), but other AI models (such as image recognition, image/video creation) are also ANNs, so it seems to make sense to me to include those.