For something to be computable, it only needs to be possible to show that it is logically possible, by e.g. decomposing the problem into elements we know are computable or showing an example.

The existence of the human brain absent any evidence of any supernatural element is strong evidence that human reasoning is computable, and it's a reasonable, testable, falsifiable hypothesis to make: If you want to counter it "all" you need to do is to show evidence of any state transition in even a single brain that does not follow known laws of physics. Just one.

Alternatively, even just coherently describing a decision-making process that it is possible to construct a proof wouldn't be computable using known logic.

Either would get you a Nobel Prize, in either physics or maths. Absent that, even just a testable hypothesis that if proven would increase the likelihood of finding either of the above would be a huge step.

In the absence of all of that, it's pure faith to presume human reasoning isn't computable.