Confusion regarding heat capacity $C_V $ for a Van der Waals gas

Suppose we have the following cycle in the $P-V$ diagram for a Van der Waals gas: The internal energy , and equation of state for a VdW gas being: $U = C_V T- a\dfrac{N^2}{V}$ $P + a\dfrac{N^2}{V^2} = \dfrac{Nk_B T}{V-Nb}$ and the heat capacities being linked by the Mayer relation: $C_P - C_V = TV \dfrac{\alpha^{2}}{\kappa_T}$ so one could calculate the coefficients $\alpha , \kappa_T$ from the equation of state. My textbook says that heat transfered from $A \rightarrow B$ is: $Q_{AB} = \Delta U - \underbrace{W_{AB}}_\text{$=0$} = \int^B_A C_V dT = C_V (T_2 - T_1)$ It can be show that $C_V$ is volume independent for a VdW gas but here we assume that $C_V$ is temperature independent , how does one justify that ? Continuing, if we calculate the heat transfered by going from $C$ to $A$ : $Q_{CA} = \Delta U - W_{AC} = \int^A_C C_P dT \neq C_P \Delta T$ but this time $C_P$ is not temperature independent , which makes sense given the various terms in the Mayer relation depend on $T$ . Summary: How does one justify $C_V$ being temperature independent for $A \rightarrow B$ ?