Of course you can build such a detection setup. The only question is what it will do.
To get a grip of things, let us consider three pairs of input states - horizontal/vertical (H/V), diagonal/anti-diagonal (45/-45 degrees, D/A), and circular left/right-handed (L/R) polarization. With H/V polarization on the HV detector, H is always detected by H detector and V by V detector, on the DA detector each gets split in half on both detectors. With D/A polarization, it is the other way round - D always ends on D detector but gets split on H/V detector. L/R light gets always split on both H/V and D/A discriminators.
More rigorously, assuming a definite polarization state of the pulse, this can be written in a superposition of H and V polarization as $|\psi\rangle=c_H|H\rangle+c_V|V\rangle$. Simultaneously, this can be written in the D/A basis as $|\psi\rangle=c_D|D\rangle+c_A|A\rangle$. Because these two bases are connected by $|D\rangle=(|H\rangle+|V\rangle)/\sqrt{2}$, $|A\rangle=(|H\rangle-|V\rangle)/\sqrt{2}$, there is a connection between coefficients in the two bases
$$ c_D+c_A = \sqrt{2}c_H $$
$$ c_D-c_A = \sqrt{2}c_V. $$
By measuring the polarization, we obtain information about the modulus squared of these coefficients, $|c_i|^2$, where $i = H,V,D,A$. If we now express $c_{D,A}$ in terms of $c_{H,V}$, we get four equations for the coefficients $c_{H,V}$ together with $c_{H,V}^\ast$. This means that we can obtain full information about $c_{H,V}$ from the measurement.
Although this scheme can work in this way, the usual approach is (as far as I know) to use only one PBS and one pair of detectors and change the basis in which they operate. This is advantageous because you get more light on each detector so the SNR is better.
