= Solution
Let $\overline{\mathbf r'}=(x',y',-z')$ be the reflection of the source in the equatorial plane. For a source in the upper half-ball, the <Dirichlet Green function inside a hemisphere> is
$$
\boxed{G_{\rm hemi}(\mathbf r,\mathbf r')=H(\mathbf r,\mathbf r')-H(\mathbf r,\overline{\mathbf r'}).}
$$
Both terms vanish on the spherical boundary. On the flat boundary $z=0$, the two source distances are equal, and the distances to their spherical image points are also equal, so the difference vanishes there too. The reflected physical source is in the lower half-ball and both inverted image points are outside the sphere. Therefore the only source singularity in the desired region is the original one, with coefficient $+1$ in $\Delta G$. This combines reflection with the spherical <method of images>.
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