Edit. 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.
Second suggestion. 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, Moduli of Curves, 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.
First attempt with ridiculous error because the odd Bernoulli numbers equal zero.
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.
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, $$\rho_*\text{Td}(\rho) = -2[X]\cap \sum_{m\geq 0} \frac{B_{2m+1}}{(2m+1)!} a^m.$$
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 $r=0,\dots,\text{dim}(B).$