The answer is `No'.

**Example:**

Let $A$ be any C*-algebra that is not finitely generated as a C*-algebra. For example, let $A=C(X)$ for a compact, metric space $X$ that has infinite covering dimension, such as the Hilbert cube.

Since $A$ is a C*-algebra, the family $\mathfrak{A}=\{A\}$, consisting just of $A$ itself, is a trivial exhausting, directed family of complete subalgebras. However, there is another such family: For a subset $F$ of $A$, let $C^*(F)$ denote the sub-C*-algebra generated by $F$. Consider the family
$$
\mathfrak{B} = \{ C^*(F) \colon F \text{ finite subset of } A\}.
$$
It is easy to check that $\mathfrak{B}$ is an exhausting, upward directed family of complete subalgebras of $A$.
However, since $A$ itself is not finitely generated, the subalgebra $A$ does not belong to the family $\mathfrak{B}$.

Now, consider the identity map $A\to A$, which is clearly a c.p.c. order zero map. When the source is equipped with the family $\mathfrak{A}$ and the target is equipped with the family $\mathfrak{B}$, then the OP's question has a negative answer.

**Explanation**

A local C*-algebra is only assumed to have an exhausting, directed family of complete subalgebras. But this family is not part of the structure of the local C*-algebra. Indeed, there may be different (inequivalent) such families for the same local C*-algebra, as the above example shows.

A different (and maybe better) way to think of local C*-algebras is as follows. Given a pre-C*-algebra $A$, consider the embedding into its completion $\overline{A}$. Then $A$ is a local C*-algebra if and only if for each finite subset $F$ of $A$, the C*-algebra $C^*(F)$ (generated in $\overline{A}$) is contained in $A$. This shows that every local C*-algebra has a canonical exhausting, directed family of complete subalgebras, namely the collection of all finitely generated sub-C*-algebras.

It seems plausible that the structure theory for c.p.c. order zero maps between C*-algebra (as developed by Winter and Zacharias), can be extended to the context of local C*-algebras to show the following:

Let $A$ and $B$ be local C*-algebras, and let $\varphi\colon A\to B$ be a c.p.c. order zero map. Let $F$ be a finite subset of $A$. Then there exists a finite subset $G$ of $B$ (presumably of cardinality at most one more than that of $F$) such that the image of the C*-algebra $C^*(F)$ under $\varphi$ is contained in the C*-algebra $C^*(G)$.