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For I am the God of science.
If you have set the experiment up so that both beams are definitely known to be vertically-polarized, I'm not sure why you think putting one of the beams through a filter would change that fact for either beam. It seems more likely that the filter would simply break the entanglement. You can't force entangled particles to be in one state or another.
Yes, you are right. I thought that if you start with a polarized beam, let it go through the BBO crystal the entangled particles will have the initial polarization. But entangled particles always have opposite spin upon measurement. I don't see how this can work.