= Solution
Amplify the chiral phase by the identity on the auxiliary $\mathbb C^2$, and let $P$ act by multiplication on each chirality. The displayed projection has image line $E_z=\mathbb C(1,\overline z)$. On the chart $w=1/z$ near infinity it has frame $(\overline w,1)$. On the equator the normalized second frame is $z$ times the first, so its clutching function has winding $+1$. Equivalently it is the conjugate of the tautological degree-$-1$ line. Thus $E$ has degree $+1$ with the complex orientation fixed in part (c).
It is equivariant for the conjugate defining action on the auxiliary $\mathbb C^2$; this representation is equivalent to the defining representation of $SU(2)$. At $z=0$ its isotropy character under $\operatorname{diag}(e^{it},e^{-it})$ is $e^{-it}$. Tensoring with the positive spinor weight $+1$ gives weight zero on $S^+\otimes E$, whereas tensoring with the negative weight $-1$ gives weight $-2$ on $S^-\otimes E$.
The compressed map is
$$
PVP:P(H_+\otimes\mathbb C^2)\longrightarrow P(H_-\otimes\mathbb C^2).
$$
Its parametrix is $PV^*P$: the two products equal the respective identity $P$ modulo compacts, since every matrix entry of $P$ is a continuous function and the phase commutators in part (c) are compact. Hence this is an equivariant <Fredholm operator>.
Use the precisely stated homogeneous-bundle multiplicity rule from part (c). The domain, of weight zero, decomposes as $E_0\oplus E_2\oplus E_4\oplus\cdots$. The target, of weight $-2$, decomposes as $E_2\oplus E_4\oplus\cdots$. Part (b) subtracts these multiplicities, leaving only the trivial representation. Therefore
$$
\boxed{\operatorname{ind}_G(PVP)=[E_0],\qquad\operatorname{ind}(PVP)=1.}
$$
This computes the index from the actual projection and representation weights without invoking an index theorem that merely supplies the requested number. Reversing the chiral direction gives $-1$. If $V$ instead denotes the full self-adjoint phase on $H_+\oplus H_-$, its compression is self-adjoint <Fredholm> and has index zero; the nonzero answer uses the standard chiral interpretation made explicit above.
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