Let me attempt an answer, but be warned that I am not usually doing this much point-set topology, so I may well have missed something. (I really hope I didn't...)
It is clear that if $X$ factors (up to homotopy) through a CW-complex of dimension $n$, then it actually factors (up to homotopy) through $\mathbb{CP}^n\hookrightarrow\mathbb{CP}^\infty$, by cellular approximation. So we might as well show this stronger claim. Combined with the classification theorem for vector bundles on paracompact Hausdorff spaces, we want to prove that a complex line bundle on a space of covering dimension $n$ can be generated by $n+1$ global sections, alternatively that there is a trivializing covering by $n+1$ open sets. (The trivializing cover provides an explicit map $X\to\mathbb{CP}^n$ classifying a line bundle isomorphic to the one corresponding to the original map $X\to\mathbb{CP}^\infty$. By the classification theorem the maps are homotopic.)
Now we apply Ostrand's theorem, cf. Theorem 3.2.4 of Engelking's book "Dimension theory", noting that paracompact Hausdorff spaces are normal:
A normal space $X$ has covering dimension $\leq n$ if and only if for every locally finite open cover $\{U_\alpha\}_{\alpha\in I}$ there exists an open cover $\{V_\beta\}$ of $X$ which can be represended as the union of $n+1$ families $\mathcal{V}_1,\dots,\mathcal{V}_{n+1}$ with $\mathcal{V}_i=\{V_{i,\alpha}\}_{\alpha\in I}$ such that $V_{i,\alpha}\cap V_{i,\alpha'}=\emptyset$ and $V_{i,\alpha}\subseteq U_\alpha$ for $\alpha\in I$, $i\in 1,\dots,n+1$.
So, take a trivialization of the line bundle $\mathcal{L}\to X$, and refine the trivializing cover to one satisfying the above property. Then for each $i\in\{1,\dots,n+1\}$, we can define a global section on $\bigcup_{\alpha} V_{i,\alpha}$ by taking non-vanishing sections on the open sets $V_{i,\alpha}$. Using a partition of unity precisely subordinate to our open cover, we can glue the local sections to a global section which is non-vanishing on $\bigcup_{\alpha}V_{i,\alpha}$ and zero outside. (If we assume that $X$ is metric, then we could alternatively scale the section on $\bigcup_{\alpha}V_{i,\alpha}$ with the distance to the boundary to get an extension of the section.)
Applying the above construction to the families $\mathcal{V}_i$ we get $n+1$ global sections generating the line bundle. Hence there is a classifying map $X\to\mathbb{CP}^\infty$, homotopic to the original one, which factors through $\mathbb{CP}^n$.
For what it's worth, let me remark that the proof for covering dimension zero is a lot easier. If the space has covering dimension zero, we can get a trivializing cover by disjoint open sets. Clearly, on such a cover we can define a global non-vanishing section of the line bundle, showing that the classifying map $f:X\to\mathbb{CP}^\infty$ is null-homotopic. This actually generalizes to show that any map from a space of covering dimension zero to any CW-complex (locally contractible suffices) is null-homotopic. It is however not clear to me how to extend this to positive dimensions - the extension of homotopies from open sets to the whole space, done for sections via the partition of unity, seems to be complicated in general.