"any desired level of precision" is actually the issue. The moment you choose a level of precision, you cease being accurate (at that level). If you make the argument that a TM has infinite memory, and can therefore represent an infinite precision, then I would counter that our current defintion of a TM requires a finite tape alphabet (and finite number of states), which is part of the TM's known computational limitations. And, of course, the moment that you use any finite set of symbols to represent an infinitely precise value, you fall into the problem that the set of real numbers has a larger cardinality than the set of possible turing machines (again, simple proof via diagonalization).

It is possible that the brain's imprecision (I would argue that "inconsistency" might be a better word) is a requirement of it's computational ability. Again, we haven't defined how the brain computes, nor do we have a model for explaining its computation, encoding or representation of knowledge, or emergent behavior. We have observed phenomena related to some of these things, but we are far from understanding it. It may be that the computational processes are dependent on the surrounding environment. We know that the biological processes are influenceable by the physical world, but we do not know much about how these external forces affect, limit, or are required for, the process of brain computation.

The quantum world may play a part in consciousness (or no, we don't know). Non-determinism may play a part. It is possible that, in order to simulate a brain, one would have to simulate the entire universe around it in order to predict the behavior... meaning that it may well require a universe to perform the simulation.

Which brings us to a related theory of whether or not we are living in a simulation, but I digress... :)