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When we know that the electric field outside a uniformly charged sphere is the field of a point charge placed at the center of the sphere, it is sufficient to use Newton's Third Law to prove that the two spheres act on each other with the same forces as two point charges. There is no need to consider integrals, this is an exercise in logic. Indeed, the first sphere acts on the second sphere with the same force as a point charge at the center of the first sphere. Consequently, the second sphere acts on the first sphere with the same force with which it acts on a point charge at the center of the first sphere. But the second sphere acts on a point charge as a point charge at its center.

Update. While I think the original version of my answer is sufficient, after the questioner's comment I decided to give a little more detail. We will consider the second charged sphere as a set of infinitesimal charges $e_i$ with coordinates $\vec{r}_i$. Then the force with which the first sphere acts on the second sphere is equal to $$ \vec{F}_{12} = \sum_i e_i \vec{E}_i^{(1)}, $$ where $\vec{E}_i^{(1)}$ is the electric field created by the first sphere at point $\vec{r}_i$. In the case when the center of the first sphere is at the coordinate center, we have $$ \vec{E}_i^{(1)} = \frac{kQ_1\vec{r}_i}{|\vec{r}_i|^3} $$ and accordingly $$ \vec{F}_{12} = Q_1\sum_i \frac{ke_i\vec{r}_i}{|\vec{r}_i|^3} = -Q_1 \vec{E}_0^{(2)}, $$ where $$ \vec{E}_0^{(2)} = -\sum_i \frac{ke_i\vec{r}_i}{|\vec{r}_i|^3} $$ is the electric field created by the second sphere at the first sphere center. In case when the second sphere center is at $\vec{R}$, we have $$ \vec{E}_0^{(2)} = -\frac{kQ_2\vec{R}}{|\vec{R}|^3} $$ After all we obtain $$ \vec{F}_{12} = \frac{kQ_1Q_2\vec{R}}{|\vec{R}|^3} $$

When we know that the electric field outside a uniformly charged sphere is the field of a point charge placed in the center of the sphere, it is sufficient to use Newton's Third Law to prove that the two spheres act on each other with the same forces as two point charges. There is no need to consider integrals, this is an exercise in logic. Indeed, the first sphere acts on the second sphere with the same force as a point charge in the center of the first sphere. Consequently, the second sphere acts on the first sphere with the same force with which it acts on a point charge in the center of the first sphere. But the second sphere acts on a point charge as a point charge in its center.

Update. After the questioner's comment I decided to give a more detailed answer. We will consider the second charged sphere as a set of infinitesimal charges $e_i$ with coordinates $\vec{r}_i$. Then the force with which the first sphere acts on the second sphere is equal to $$ \vec{F}_{12} = \sum_i e_i \vec{E}_i^{(1)}, $$ where $\vec{E}_i^{(1)}$ is the electric field created by the first sphere at the point $\vec{r}_i$. In the case when the center of the first sphere is in the coordinate center, we have $$ \vec{E}_i^{(1)} = \frac{kQ_1\vec{r}_i}{|\vec{r}_i|^3} $$ and accordingly $$ \vec{F}_{12} = Q_1\sum_i \frac{ke_i\vec{r}_i}{|\vec{r}_i|^3} = -Q_1 \vec{E}_0^{(2)}, $$ where $$ \vec{E}_0^{(2)} = -\sum_i \frac{ke_i\vec{r}_i}{|\vec{r}_i|^3} $$ is the electric field created by the second sphere in the center of the first sphere. In the case when the center of the second sphere is at the point $\vec{R}$, we have $$ \vec{E}_0^{(2)} = -\frac{kQ_2\vec{R}}{|\vec{R}|^3} $$ After all we obtain $$ \vec{F}_{12} = \frac{kQ_1Q_2\vec{R}}{|\vec{R}|^3} $$

When we know that the electric field outside a uniformly charged sphere is the field of a point charge placed at the center of the sphere, it is sufficient to use Newton's Third Law to prove that the two spheres act on each other with the same forces as two point charges. There is no need to consider integrals, this is an exercise in logic. Indeed, the first sphere acts on the second sphere with the same force as a point charge at the center of the first sphere. Consequently, the second sphere acts on the first sphere with the same force with which it acts on a point charge at the center of the first sphere. But the second sphere acts on a point charge as a point charge at its center.

When we know that the electric field outside a uniformly charged sphere is the field of a point charge placed at the center of the sphere, it is sufficient to use Newton's Third Law to prove that the two spheres act on each other with the same forces as two point charges. There is no need to consider integrals, this is an exercise in logic. Indeed, the first sphere acts on the second sphere with the same force as a point charge at the center of the first sphere. Consequently, the second sphere acts on the first sphere with the same force with which it acts on a point charge at the center of the first sphere. But the second sphere acts on a point charge as a point charge at its center.

Update. While I think the original version of my answer is sufficient, after the questioner's comment I decided to give a little more detail. We will consider the second charged sphere as a set of infinitesimal charges $e_i$ with coordinates $\vec{r}_i$. Then the force with which the first sphere acts on the second sphere is equal to $$ \vec{F}_{12} = \sum_i e_i \vec{E}_i^{(1)}, $$ where $\vec{E}_i^{(1)}$ is the electric field created by the first sphere at point $\vec{r}_i$. In the case when the center of the first sphere is at the coordinate center, we have $$ \vec{E}_i^{(1)} = \frac{kQ_1\vec{r}_i}{|\vec{r}_i|^3} $$ and accordingly $$ \vec{F}_{12} = Q_1\sum_i \frac{ke_i\vec{r}_i}{|\vec{r}_i|^3} = -Q_1 \vec{E}_0^{(2)}, $$ where $$ \vec{E}_0^{(2)} = -\sum_i \frac{ke_i\vec{r}_i}{|\vec{r}_i|^3} $$ is the electric field created by the second sphere at the first sphere center. In case when the second sphere center is at $\vec{R}$, we have $$ \vec{E}_0^{(2)} = -\frac{kQ_2\vec{R}}{|\vec{R}|^3} $$ After all we obtain $$ \vec{F}_{12} = \frac{kQ_1Q_2\vec{R}}{|\vec{R}|^3} $$

When we know that the electric field outside a uniformly charged sphere is the field of a point charge placed at the center of the sphere, it is sufficient to use Newton's Third Law to prove that the two spheres act on each other with the same forces as two point charges. There is no need to consider integrals, this is an exercise in logic. Indeed, the first sphere acts on the second with the same force as a point charge at the center of the first sphere. Consequently, the second sphere acts on the first sphere with the same force with which it acts on a point charge at the center of the first sphere. But the second sphere acts on a point charge as a point charge at its center.

When we know that the electric field outside a uniformly charged sphere is the field of a point charge placed at the center of the sphere, it is sufficient to use Newton's Third Law to prove that the two spheres act on each other with the same forces as two point charges. There is no need to consider integrals, this is an exercise in logic. Indeed, the first sphere acts on the second sphere with the same force as a point charge at the center of the first sphere. Consequently, the second sphere acts on the first sphere with the same force with which it acts on a point charge at the center of the first sphere. But the second sphere acts on a point charge as a point charge at its center.