Electric field from the intersection of two spheres with different charges

This question is repeated a lot, but I for two spheres with different charge densities, sphere one with radius a centered at the origin and charge density $$\rho_1$$, and sphere two with radius centered at a point d on the x axis a and charge density $$\rho_2$$, and their intersection is has a charge density $$\rho_1+\rho_2$$ i.e:

\begin{align} \rho(\vec{r}) &=\rho_1 \;\;\;\;\;\;\;\;\text{for}\; |\vec{r}| \leq a \; \text{and} \; |\vec{r}-d \hat{e_x}| > a \\ &=\rho_2 \;\;\;\;\;\;\;\;\text{for}\; |\vec{r}| > a \; \text{and} \; |\vec{r}-d \hat{e_x}| \leq a \\ &=\rho_1+\rho_2 \;\text{for}\; |\vec{r}| \leq a \; \text{and} \; |\vec{r}-d \hat{e_x}| \leq a \\ &=0 \;\;\; \text{else} \end{align}

My idea was to assume that both spheres have the charge $$\rho_1+\rho_2$$, calculate the electric field using Gauss Law for both sphere, and then use super position principle to find the field inside the intersection, anyhow I am not completely sure if I am allwoed to use this assumption that both spheres are charged with $$\rho_1+\rho_2$$, so can I make this assumption? or should I assume that each sphere is only charges with $$\rho_1$$ or $$\rho_2$$ respectively and then use the superposition principle?

Yes, you can assume that each sphere is only charged with $$ρ_1$$ and $$ρ_2$$ respectively and then use the superposition principle (justification below).
So, in this case, you can look at this problem as if you have 2 distinct charged spheres with $$\rho_1$$ and $$\rho_2$$. Now, the amazing thing is, even though these two have an intersection, thanks to superposition, we can split the current density in that region ($$\rho_1 + \rho_2$$) as one region with $$\rho_1$$ and one region with $$\rho_2$$. Then you can think the $$\rho_1$$ part along with the rest of $$\rho_1$$ as a full sphere, and the same for $$\rho_2$$.