To see why I find your argument unconvincing, replace 'consciousness' with 'transistors', and 'ToE' with 'quantum theory'.

"If we can understand transistors after constructing quantum theory, then we should be able to understand transistors after constructing systems which are similar enough to quantum theory but do not in fact correspond to reality. If you can agree with this statement,"

I do,

"then you might also agree that If we can understand transistors after constructing quantum theory, then there is in fact a space of mathematical systems which would provide an adequate framework to understand transistors"

There is. But this says nothing, because even though people could have understood transistors before quantum theory, there would have been many competing hypotheses but no prior to help them sort out the hypotheses. Quantum physics provided a prior which allowed people to selectively narrow down hypotheses.

"Does it not follow that we should theoretically be able to construct a mathematical system within which transistors could exist?"

We could, but there would be so many mathematical systems with this property that finding the correct one would be hopeless.