= Solution
A full fringe scan is unnecessary. Use the same components to make three fixed <projective measurements>, on separate batches of identically prepared <photons>. Direct path detection after the <polarizing beam splitter> measures the $H/V$ populations. Recombine after matching the arm <photon polarizations>, once with $\theta=0$ and once with $\theta=\pi/2$, to measure the two quadratures of the <quantum coherence>.
In the $H/V$ basis, write the <density matrix> in terms of its <Bloch vector> as $\rho=(I+r_xX+r_yY+r_zZ)/2$. The previous formula becomes
$$
P_1(\theta)=\frac12+\frac12(r_x\cos\theta-r_y\sin\theta).
$$
Thus three measured relative frequencies give
$$
\boxed{r_z=P(H)-P(V),\qquad r_x=2P_1(0)-1,\qquad r_y=1-2P_1(\pi/2),}
$$
and determine
$$
\boxed{\rho=\frac12\begin{pmatrix}1+r_z&r_x-ir_y\\r_x+ir_y&1-r_z\end{pmatrix}.}
$$
The two interferometric settings analyze the diagonal <linear polarizations> $(|H\rangle\pm|V\rangle)/\sqrt2$ and the <circular polarizations> $(|H\rangle\pm i|V\rangle)/\sqrt2$, with detector labels fixed by the calibrated <quantum phase> convention. This is <informationally complete three-observable qubit tomography>: direct measurements in three bases replace fitting an entire phase-dependent fringe. It also reconstructs a mixed input, for which $|r|\le1$, although the source here is assumed pure.
Simply measuring transmission through one fixed <polarizing beam splitter> would determine only $r_z$. Rotating a linear analyzer through different angles can additionally determine $r_x$, but cannot distinguish the two signs of $r_y$ without a relative-phase measurement. For arbitrary, possibly elliptical <photon polarization>, all three settings above are needed; no extra analyzer element beyond the described path-phase control and <photon polarization> rotator is required.
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