I am glad that by putting our heads together in Munster this week we finally have a proof and can answer this question (a full 2 years after it was asked!). The answer is yes, and I want to sketch here the email I sent you in case it might be of use to others who are trying to prove things are cellular. Let Top mean compactly generated spaces. Let I (and J) denote the generating (trivial) cofibrations of Top$_\ast$. Let W(I) (and W(J)) denote the generating (trivial) cofibrations of your category of functors, so these sets are generated by functors of the form CW$_\ast(X,-) \wedge i^n_+$ where $i^n_+$ runs through the maps in I (resp. J). We have three conditions to check.
(1) We must show the domains of maps in W(J) are small (for some $\kappa$) relative to cofibrations in Fun(CW$_\ast$,Top$_\ast$). This is easy, because Top is cellular, so domains of maps in J are small relative to cofibrations and the representable functors CW$_\ast(X,-)$ preserve filtered colimits (since finite CW complexes are compact, i.e. small relative to $\kappa = \aleph_0$). So smallness of domains of J relative to I-cell implies smallness of domains of W(J) relative to WI-cell. An easier way to see this is to note that maps in W(J) are objectwise inclusions, colimits are computed objectwise in Fun(CW$_\ast$, Top$_\ast$), and all spaces are small relative to inclusions.
(2) We must show cofibrations are effective monomorphisms. We know the generators are objectwise closed inclusions, i.e. for all $Y$, $CW_\ast(X,Y)\wedge i_\ast^n$ is a closed inclusion, since it's a compact space smashed with a closed inclusion. But in Top, closed inclusions are effective monomorphisms (indeed, these two classes coincide!) so we're done.
(3) We must show the domains and codomains of $W(I)$ are compact in the sense of Hirschhorn relative to $W(I)$. This is somewhat painful. It means that there is some regular cardinal $\kappa$ such that for every relative $W(I)$-cell complex $\eta:F\to T$ (a map in our category, hence a natural transformation) and for every presentation of $\eta$ by a chosen collection of cells then every natural transformation $CW(X,-)\wedge K \to T$, where $K$ is a sphere or disk, factors through a subcomplex of size at most $\kappa$. A presentation of $\eta$ is a realization of $\eta$ as the colimit of a $\lambda$-sequence of maps which are pushouts of coproducts of cells (i.e. maps in $W(I)$). A subcomplex of the given presentation of $\eta$ is a $\lambda$-sequence formed by pushouts of coproducts of a subset of cells. The size is the cardinality of the set of cells. For us $\kappa$ will be taken to be larger than $\gamma_+$ where $\gamma$ is the cardinal making Top cellular. We'll follow the roadmap given by Hovey's proof of Proposition A.8 in his paper on Spectra and Symmetric Spectra in General Model Categories. In particular, we'll use his reduction that it is sufficient to consider cells which are transfinite compositions of pushouts of maps in $W(I)$ rather than coproducts of maps in $W(I)$.
The idea is to reduce to considering maps in Top, use the cellularity of Top to identify which cells we need in our sub-presentation of $T$, then prove that $\eta$ actually factors through that sub-presentation. First, evaluate $\eta$ on a given $Y$ and get $\eta_Y:CW(X,Y)\wedge K \to T(Y)$. Be careful here: it is NOT true that CW(X,Y) is cofibrant (see e.g. this MO question). However, with the compact open topology these CW(X,Y), are compact (this follows from Ascoli's theorem, for example), hence so are the spaces $CW(X,Y)\wedge K$, since K is also compact.
Write $T(Y)$ as the filtered colimit of the cells that make up $T$, evaluated at $Y$, e.g. as a transfinite composition of pushouts of maps $C_{\beta}(Y)\to D_{\beta}(Y)$. Note that Hovey demonstrates it's enough to consider such pushouts rather than pushouts of coproducts of maps in $W(I)$. The filtered colimit building $T(Y)$ consists of closed inclusions because cofibrations in Fun(CW$_\ast$,Top$_\ast$) are objectwise closed inclusions. Since we're working in compactly generated spaces, closed inclusions are automatically closed $T_1$-inclusions, so Hovey's 2.4.2 tells us that maps into such a colimit $T(Y)$ from a compact space factors through a finite subcomplex. This implies only finitely many cells have been used.
Use Hovey's method of writing the $\lambda$-sequence as a retract of another $\lambda$-sequence with the same cells (technically, a set of cells in bijection with your cells), but now formed out of maps in $I$ rather than maps in $W(I)$ evaluated at $Y$. This relies on Hirschhorn 10.5.25 (beware: Hovey was referencing a different draft of Hirschhorn's book than the published one, so his numbering does not match Hirschhorn's), but at the end allows you to apply cellularity in Top to find a subcomplex with lesser cardinality through which $\eta_Y$ factors, though verifying this carefully requires a transfinite induction following the same steps as in Proposition A.8. Let $S_Y$ be the set of $W(I)$ cells which appear (evaluated at $Y$) in the subcomplex for $\eta_Y$.
Next, form a subcomplex $R$ of $T$ which contains all the cells $S_Y$. We know this is a subcomplex, because for each $Y$ there were fewer than $\kappa$ many cells and only had the cardinality of $CW$ many Y’s, which is much less than $\kappa$. Next, use that $R(Y)$ contains all the cells in $S_Y$ so contains the subcomplex through which $CW(X,Y)\wedge K \to T(Y)$ factored through. This provides a map of functors $CW(X,-) \wedge K \to R$, though we don't automatically know it's a natural transformation. We can make it natural by "going further along if necessary", which I now make rigorous. Let $f:Y \to Z$ in CW, i.e. $f$ is a cellular map. We have to check that the naturality square commutes in Top (i.e. that $\eta_Y \circ CW(X,f) \wedge K = R(f) \circ \eta_X$). Because we started with a natural transformation, we know that the requisite naturality diagram commutes eventually in the colimit defining $T$ (i.e. replacing $R$ by $T$ above). Because f has a bounded number of cells (finite even), we can make this occur in $R$ by adding in finitely many more cells to $R$ from our presentation of $T$ (using here that we know the square will commute eventually). Even if you have to add these finitely many more cells for each map f, you still won’t reach the regular cardinal $\kappa$, so $R$ will still be a subcomplex and by construction will admit a natural transformation from $CW(X,-)\wedge K$ factoring $\eta$. This proves the domains are compact as required.
Incidentally, I think this proof method can be used to prove something more general, namely a statement of the form: "if $F:\mathcal{M} \leftrightarrows \mathcal{N}:G$ is an adjunction through which the model structure on $\mathcal{M}$ is transferred to $\mathcal{N}$, and if the model structure on $\mathcal{M}$ is cellular, and if $F$ and $G$ behave sufficiently well with respect to filtered colimits, then $\mathcal{N}$ is cellular.'' I'll try to write this up formally and fold it into a future paper.
