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.