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Sorry Lev, My answer got lost in a parallel Universe (probably the bohmian demon).
I wanted only to say that entanglement will change all. If you use an additional BS then you are indeed in the regime that I am mentioning. But the main problem is now that Bohmian mechanics is strongly sensitive to contexts so that the time ordering of the operations and the way you are doing entanglement will make your dynamic even more counter-classical. In some case Alice will receive the particle coming from her mirror (like in your example) but in other the particle could come from Bob mirror. You mentioned already these problems in your cat paper years ago and Hardy did a lot in that direction as well. Bell was probably the first to see that of course.