The correct starting point to answer this question is to recall that the definition of canonical and grand-canonical ensembles is fixed.
In the grand-canonical ensemble for a simple system, the independent variables are $T$, $V$, and $\mu$ (temperature, volume, and chemical potential), or any equivalent choice (in many cases the activity $z=\exp \left(\frac{\mu}{k_B T} \right) $ is a more convenient choice). In the canonical ensemble, the independent variables are $T$, $V$, and $N$.
The two ensembles provide the same thermodynamic information only at the thermodynamic limit, in practice, for macroscopic systems. At any finite size, the results are not the same. More important for the present question, if in a grand-canonical ensemble we select a subset of the microscopic states, according to the criterion of a fixed number of particles, we are using only part of the microscopic states. The resulting ensemble cannot describe an equilibrium thermodynamic system anymore at fixed $T$, $V$, and $\mu$ but it may be interpreted as a constrained system, where, in addition to fixing the thermodynamic variables, also the number of particles in the volume $V$ is fixed. In general, this constrained system will not provide the same thermodynamics as the unconstrained, where the average number of particles is not fixed arbitrarily but is the outcome of averaging over all the microscopic states.
The second of your formulas is not fully correct. The $N$ on the left-hand side of the equation should be intended as $\langle N \rangle$, the average number of particles. This should be clear once one recalls that the Fermi distribution $n_F$ represents the average occupation of a single state. Therefore, it is not correct to say that we fix $N$. The formula provides the average number of particles in the volume $V$ at fixed $T$ and $\mu$.
The reason for introducing the expression for $\langle N \rangle$ is that the object we want to extract, the constant-volume specific heat, is usually given as a function of the density and not of the chemical potential. For that, the canonical ensemble would be more appropriate. Still, due to the constraints induced by Pauli's principle, calculations for an ideal Fermi gas are easier in the grand-canonical ensemble.