It is obvious the flux of the whole sphere is $\iiint_V 3r^2 dV = \frac{12}5\pi R^5 = 7500\pi$. I guess there's no problem you getting this.
Then we subtract the flux in the region with $|x|,|y|,|z|\ge 4$ from $7500\pi$. We notice that this will create 6 circular holes from the sphere. By symmetry the flux of all 6 holes should be the same, so we only need to compute one.
I believe there is no other way than computing $\iint_S \mathbf F\cdot \hat{\mathbf n}\,dS$ directly. This method does not really use divergence theorem, but I doubt there is any simpler methods.
The extra cap can be described in spherical coordinates as $\left\{r=5\wedge\theta \in \left[0, \tan^{-1}\frac34\right)\right\}$. The vector field can be rewritten as
\begin{align} \mathbf F &= x^3\hat{\mathbf x} + y^3\hat{\mathbf y} + z^3 \hat{\mathbf z} \\ &= (r\sin\theta\cos\phi)^3 (\sin\theta\cos\phi \hat{\mathbf r}+\dotsb) + (r\sin\theta\sin\phi)^3 (\sin\theta\sin\phi \hat{\mathbf r}+\dotsb) + (r\cos\theta)^3 (\cos\theta \hat{\mathbf r}+\dotsb) \\ &= r^3 \left( \sin^4\theta(\cos^4\phi + \sin^4\phi) + \cos^4\theta \right) \hat{\mathbf r} + \dotsb, \end{align} because $\hat{\mathbf n}dS=r^2 \sin\theta\, d\phi \, d\theta\hat{\mathbf r}$ on the spherical surface, we are left with \begin{align} \iint_S \mathbf F\cdot \hat{\mathbf n} \,dS &= \int_0^{2\pi}\int_0^{\tan^{-1}(3/4)} r^3 \left( \sin^4\theta(\cos^4\phi + \sin^4\phi) + \cos^4\theta \right) \cdot r^2 \sin\theta\, d\theta \, d\phi \\ &= 5^5 \int_0^{\tan^{-1}(3/4)}\int_0^{2\pi} \left( \sin^5\theta(\cos^4\phi + \sin^4\phi) + \cos^4\theta\sin\theta \right) d\phi \, d\theta \\ &= 5^5\pi \int_0^{\tan^{-1}(3/4)} \left( \frac32 \sin^5\theta + 2\cos^4\theta\sin\theta\right)d\theta \\ &= 5^5\pi \left( \frac32 \cdot \frac{428}{46875} + 2\cdot \frac{2101}{15625} \right) \\ &= \frac{4416}5 \pi, \end{align} hence the final answer is $ 7500\pi - 6\times\frac{4416}5 \pi = \frac{11004}5\pi \approx 6914.0171. $
As "verification", I did arrive at the same answer by numerical integration.
You should need to show yourself how to carry out the integrals of $\sin^5\theta$ etc.