Does the beamsplitter have to be non-polarizing? My understanding is that standard beamsplitters do the splitting by polarizing the incoming light.

Now that I think of it, you're probably using a standard display as your image source? That does indeed produce polarized light. In that case I would try a (common in photography) circular polarizing filter. Rotate the filter to match the display polarization, after that the light will be circularly polarized. I think you could use that in combination with a "standard" polarizing beamsplitter. Depending on the desired dimensions, there are a few available on aliexpress.