Yeah, a proper refutation of the idea that cognition is computation is going to have a VERY difficult time addressing the fact that computation and information theory are VERY fundamental physics, right down there with entropy, thermodynamics, and causality.
I just don't see how you can possibly get around that without figuratively introducing fairies. More susinctly: if it's not that, I think it's going to involve a proposal that involves something outside of observed physics which puts it on very dubious ground.
If you don't accept that, you have some really fundamental truths you're going to have to explain around.
1. By all measures, cognition is intimately associated with the physical brain (and the fundamental forces, chemistry, and emergent phenomenon of a functioning brain). This is basically irrefutable by every measurement and understanding of physics that we have. We can even use drugs, strong electric or magnetic fields, or physical stimulation to interrupt and alter that cognition.
2. Computation itself is such a fundamental notion involving information and operations on that information that most emergent phenomenon you could propose will almost necessarily meet the definition of computation.
Now... exactly what TYPE of computation is an interesting question: is it equivalent to a turing machine? To this, I don't know, but I suspect so. There are many other viable computational models though that likely fit the bill if this fails.