I was doing some further research and an AC controlled filter called "Fast Polarization Modulator" does this kind of polarization switching that uses LC layer and polarizing filter like in LCD panels. For example a company in Sweden called LC-Tec (www.lc-tec.se) sells such modulators of proper size (2x2") for my application, which has ~44% transmittance. The basic FPM is able to do reasonably fast transition between the polarization stages (150us to non-twisted stage, 35ms to 90 degree twisted stage), while X-FOS of LC-Tec does faster 50us vs 1.6ms switching for applications requiring higher speed switching. The main issue for small-budget projects is the price however as the basic FPM retails for about 300USD.
As some people mentioned in comments, another potential option is to use polarizing beamsplitter that splits the light to P (transmitted) and S (90 degree reflected) polarized light. Then a first/front surface mirror (to avoid ghosting vs second surface mirror and further polarization) could be used to direct reflected S-polarized light back towards the camera lens. This would effectively result as a single image being split into two, where one half of the image is S-polarized version and the other half P-polarized version of the same image. The advantages are that this is completely passive system requiring no active control mechanism and thus doesn't have any limit for switching speed either. Also polarizing beamsplitter are able to retain around 90% of the transmitted & reflected light vs 44% of FPM. The disadvantage is that the resolution of the image is halved and there may also be more constraint regarding angle of incidence (i.e. at the edges of the image there might be some polarization discrepancies). Broadband Polarizing Plate Beamsplitter with AR coating (to reduce ghosting) of size 25x25mm retails for 199USD at Edmund Optics (www.edmundoptics.com) though, so it's quite expensive piece for projects with small budgets. This size might be too small as well for this particular application. FS-mirrors of proper size for this application with ~97% reflectance for visible light range can be bought for about 20USD from AliExpress (www.aliexpress.com).
If the cost of polarizing beamsplitter is too high, another alternative is to use non-polarizing 50/50 beamsplitter and adding super high contrast polarizing filters both behind the beamsplitter and between camera lens and FS-mirror. The downside with this cheaper solution is that only ~44% of the light gets transmitted through the system (vs ~90% of using polarizing beamsplitter). Non-polarizing 50/50 BS of proper size for this application can be bought for around 25USD at AliExpress. Polarization.com sells a 25x15cm sheet of "Super High Contrast Polarizer" for 22USD.