Unfortunately there is no such filtration.

At first, this looks very similar (but not as strong as) asking for a map of $E_\infty$ spaces $\coprod BU(n) \to BU$ which would become a splitting map $ku \to bu$ of spectra. We know that doesn't happen, because there's a nontrivial $k$-invariant.

We can look more carefully at how this $k$-invariant works and it leads us to a method to give an actual contradiction. The $k$-invariant is detected by the fact that the generator $1_{ku}$ of $\pi_0 ku$ is annihilated by the Hopf element $\eta$, and the Toda bracket $\langle 2, \eta, 1_{ku}\rangle$ contains the Bott element $\beta$ and does not contain zero. This is spelled out in the following way on the level of $E_\infty$ spaces. Suppose $X$ is an $E_\infty$ space with multiplication $\smile$ and associated spectrum $KX$, and let $\alpha \in \pi_0(X)$ have image $[\alpha] \in \pi_0(KX)$. Up to translating path components, the element $\eta[\alpha] \in \pi_1(KX)$ lifts (up to changing path components) to the element $\alpha \smile_1 \alpha \in \pi_1(X, \alpha \smile \alpha)$. We can find a canonical nullhomotopy of the path composite of $\alpha \smile_1 \alpha$ with itself, expressing the identity $2 \eta [\alpha] = 0$; if we also have a nullhomotopy of $\alpha \smile_1 \alpha$, then we can use this to construct a representative for the bracket. If you carry this out for the basepoint of $BU(1)$ using the standard map $E\Sigma_2 \times_{\Sigma_2} BU(1)^2 \to BU(2)$, you find that you get the generator of $\pi_2 BU(2) = \Bbb Z$. However, in $bu$ this would mean that there was a bracket $\langle 2,\eta, 0\rangle$ which did not contain zero; that would be bad.

Of course, it turns out to be a lot easier to appeal to some machinery. Kochman calculated the Dyer-Lashof operations on $H_* BU = \Bbb F_2[x_1, x_2, \dots]$. His calculations show, for example, that $Q^4 x_1 = x_1^3 + x_1 x_2 + x_3$. As a result, the map $H_6 (E\Sigma_2 \times_{\Sigma_2} BU(1)^2) \to H_6(BU)$ is surjective, but the map $H_6(BU(2)) \to BU$ is not (it is missing $x_1^3$). This means that our operad structure map $P(2) \times X_1 \times X_1 \to X$ could never land in a subobject $X_2$ as you want. (This assumes that I've read and understood correctly in calculating the operations.)

(I actually find Priddy's method for calculating the Dyer-Lashof operations here a little easier than trying to understand Kochman's algorithm: Priddy calculates the Dyer-Lashof operations on $H_*(\coprod BU(n)) = \Bbb F_2[a_0,a_1,\dots]$ and then you can deduce the operations on the generators $x_i = a_i a_0^{-1}$ of $H_* BU$ by the Cartan formula. The inverse screws up the property of preserving homogeneous degree and it's what's messing us up here.)