Computability of equality to zero for a simple language
This is a rather tricky question! As you seem to understand, the real issue is the presence of $\hat{}$. It is intimately related to a well known conjecture: Schanuel's conjecture, which states that, essentially, there are no non-trivial algebraic relationships between $\pi$ and $e$.

The (expected) positive answer to this conjecture would give you a decision procedure for comparison with $0$:


Reduce the expression using algebraic equalities (e.g. $(x^y)^z = x^{yz}$ etc.)
Separate the terms with exponents and the terms without.
Check that the base ($x$ in $x^y$) of the terms with exponents are all linearly independent.
The term is 0 iff it is trivial i.e. it reduces to 0 by algebraic operations.


A related question is Tarski's exponential function problem which involves variables. However it is relatively simple to reduce concrete expressions with exponents to functions with variables given the Lindemann-Wierstrass theorem.

Edit: I don't know anything about the actual complexity of the problem, though. Note that it strongly depends on your computational model e.g. whether arithmetic operations constant time.

Edit 2: I made a crucial mistake: it's not the base $x$ in $x^y$ that needs to be taken but rather $y\ln(x)$, which is much harder to check for linear independence.