<B>Edit.</B> The odd Bernoulli numbers are zero, of course!  So the first attempt below is wrong.  The second suggestion uses complete subvarietes of moduli spaces of curves.

<B>Second suggestion.</B>  For every integer $n\geq 0$, there exists an integer $g(n)\geq 0$ and a projective, $n$-dimensional, closed subvariety $B$ of the moduli space $M_g$ of smooth, projective, genus $g$ curves (over $\mathbb{C}$, for simplicity).  When I have more time, I will try to add a description of these subvarieties (I believe the original construction is due to Kodaira).  Eventually we will assume that $n$ is odd, $n=2s-1$.

Up to base change of $B$ by a finite cover, this arises from a family of curves, $$\rho:C\to B.$$  The class in K-theory of $R\rho_*\mathcal{O}_C$ equals $1-[\rho_*\omega_\rho]$.  Thus, it suffices to show that the Chern character of the Hodge bundle, $\textbf{E}_\rho=\rho_*\omega_\rho$, has nonzero term in degree $n$.
Mumford computed the Chern character of the Hodge bundle on the compactified moduli space $\overline{M}_g$.  When restricted to the interior, Mumford's formula says that the $r^{\text{th}}$ graded piece of the Chern character is zero if $r$ is even, and for $r=2s-1$ odd, it equals $$\text{ch}_{2s-1}(\textbf{E}_\rho)= (-1)^{s+1}\frac{|B_{2s}|}{(2s)!}\cdot \kappa_{2s-1}.$$ If we choose $n=2s-1$ to be odd, then we are reduced to proving that the top intersection class of $c_1(\omega_\rho)$ is nonzero.  By Theorem 6.33 of Harris-Morrison, <i>Moduli of Curves</i>, the divisor class $c_1(\omega_\rho)$ is ample on $C$ (I will try to find the original source of this theorem later).  Thus, the top intersection class is positive.  Therefore $\text{ch}_{2s-1}(\rho_*\mathcal{O}_C)$ is nonzero.

It is straightforward to get a similar result for a family of surfaces $\pi:S\to B$ by choosing $S$ to be $C\times D_0$ for a fixed curve $D_0$ and choosing $\pi$ to be the composition of $\rho$ with projection onto $C$.  Perhaps by choosing $S$ to be a fiber product of $\rho$ with a family of elliptic curves, it is possible to shift the nonzero Chern classes from odd degrees to even degrees.


<B>First attempt with ridiculous error because the odd Bernoulli numbers equal zero.</B>
<del>You can create many such examples with isotrivial families.  The simplest isotrivial morphism is a smooth, projective morphism of relative dimension $1$ whose geometric fibers are isomorphic to the projective line, $$\rho:C\to X.$$  The pushforward of $T_\rho =\textit{Hom}_{\mathcal{O}_C}(\Omega_\rho,\mathcal{O}_C)$ is a locally free $\mathcal{O}_X$-module of rank $3$ whose total Chern class equals $1-a$ for a cycle class $a$ of degree $2$.  For a locally free $\mathcal{O}_X$-module $E$ of rank $2$, if $C/X$ represents the functor on $X$-schemes of invertible quotients of the pullback of $E$, then the class $a$ equals $$a = c_1(E)^2 - 4c_2(E).$$  Up to scaling and torsion, this is the unique polynomial expression in Chern classes of $E$ of cohomological degree $2$ that is compatible with arbitrary pullback and that is preserved by the operation of tensoring $E$ with invertible sheaves.</del>

<del>The first Chern class $c = c_1(T_\rho)$ has square equal to a pullback, $$c^2 = \rho^* a.$$  Thus, the relative Todd class of $\rho$ equals $$\text{Td}(\rho) = [C]\cap\left(\sum_{m\geq 0} \frac{B_{2m}}{(2m)!}\rho^*(a^m) - \sum_{m\geq 0}\frac{B_{2m+1}}{(2m+1)!}c\cup \rho^*(a^m) \right). $$  Thus, the pushforward of the Todd class equals, $$\rho_*\text{Td}(\rho) = -2[X]\cap \sum_{m\geq 0} \frac{B_{2m+1}}{(2m+1)!} a^m.$$</del>

<del>In particular, for a second smooth, projective morphism of relative dimension $1$, $$\sigma:X\to B,$$ for $S$ defined to be $C$, and for the composition $\pi=\sigma\circ \rho$, the pushforward of the relative Todd class equals $$\pi_*\text{Td}(\pi) = \sigma_*\left(-2\text{Td}(\sigma)\cap \sum_{m\geq 0}\frac{B_{2m+1}}{(2m+1)!} a^m \right).$$  In the very simplest case that $X$ equals $\mathbb{P}^1_B$, $\sigma$ is the natural projection, and for any choice of constant cross-section $s:B\to \mathbb{P}^1_B$ (the $\infty$-section, for instance), this gives $$\pi_*\text{Td}(\pi) = -[B]\cap \left(\sum_{m\geq 0} \frac{B_{2m+1}}{(2m+1)!} (s^*a)^m +2\sum_{m\geq 0} \frac{B_{2m+1}}{(2m+1)!}\sigma_*(a^m) \right).$$  To be very explict, if $E$ is the locally free sheaf $\sigma^*\mathcal{L}\oplus \mathcal{O}_{\mathbb{P}^1}(1)$ for some choice of invertible $\mathcal{O}_B$-module $\mathcal{L}$ with first Chern class $c_1(\mathcal{L})=\lambda$, this evaluates to $$\pi_*\text{Td}(\pi) = -[B]\cap \left( \sum_{m\geq 0} \frac{B_{2m+1}}{(2m+1)!}\lambda^{2m} + 4\sum_{m\geq 1} \frac{B_{2m+1}}{(2m+1)!} m\lambda^{2m-1}\right).$$  If $\mathcal{L}$ is ample, then $\lambda^r$ is nonzero for every integer $r=0,\dots,\text{dim}(B).$</del>