Another thing to try might be conditioning on the X(t) path.
Related to that, I suggest you revisit the 1D case, where [$]X_t \equiv x_0[$], a fixed real number and the process is
[$] dY_t = x_0 (1-Y_t) Y_t dt + x_0 (1-Y_t) Y_t dB_t[$].
So, the ultimate terminal density ( admitting distributions
) is [$]p(y; x_0)[$]. Now, what exactly
do you think [$]p(y; x_0)[$] is? Based on some things you wrote before, I suspect we disagree on the answer to that. Also, do you have a proof for your answer to this case?
I revisit that because I think you have to totally nail down that case before conditioning on an X-path.
Then, the next case (under a conditioning approach) would be: stick with the 1D case, but now take [$]X_t \equiv x(t)[$], a given deterministic
function of time.
So, now the 1D problem is:
[$] dY_t = x(t) (1-Y_t) Y_t dt + x(t) (1-Y_t) Y_t dB_t[$],
and the terminal density is some [$]p(y; x(\cdot))[$].
Now, can you conjecture [$]p(y; x(\cdot))[$]? Suppose [$]x(t) = \sin t[$], or something else that oscillates at arbitrary large t?
I don't know the answer -- I am just making some suggestions about an approach.