That was simply (⟨φ|⊗⟨Ψ|) | (|B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩) without doing mod square, I did it and got something that's still wrong. I believe the mistake is in how I expanded the tensor products.
The top half is my computation of ⟨φ|⊗⟨Ψ|.
The bottom half if my computation of |B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩, note there is a common factor of ((1+delta^2)^(-1/2))/sqrt(2) which I don't show in the computation, but I do later use it for the computation of |(⟨φ|⊗⟨Ψ|) | (|B_00⟩⟨B_00|) (|φ⟩⊗|Ψ⟩)|^2 correctly.
