I am using the second definition of a trace in the accepted answer of Evaluation of a trace - how does it depend on the inner product being used? but I am concerned with general Hilbert spaces (not necessarily finite)
First let's supposed $A$ has a trace (is trace class). Would using a different scalar product (which effectively means a different Hilbert space, since the scalar product is part of the definition of a Hilbert space) change the value ?
As trace class operators (and trace) can be defined as nuclear operators (and thus trace) without recourse to the inner product, trace is independent of the choice of inner product, as long as the underlying Banach space stays the same.
To be explicitly clear and avoid potential confusion: Scaling the norm should not have an effect, as the norm enters equally on the domain and codomain (which coincide in the case of the question) - see the definition in the link - and the operator is linear.