Is there a section to the genus 2 Torelli map?

The Torelli map, which maps a smooth curve to its principally polarized Jacobian, defines a morphism of algebraic stacks $$\mathscr{M}_g \longrightarrow \mathscr A_g$$ between the moduli spaces of smooth, genus $g$ curves and dimension $g$ principally polarized abelian varieties respectively. This is an immersion, which extends to a proper morphism (no longer an immersion) $$\mathscr{M}_g^{\mathrm{cpt}} \longrightarrow \mathscr A_g$$ from the space of curves of compact type (those whose Jacobian is proper). My question can be summarized by: for $g=2$, does this morphism admit a section?

Here is further context. In paragraph 6 of de Jong and Oort's paper "On extending families of curves" is defined a principally polarized abelian (ppa) scheme $f:Y \to \mathbb{P}_k^1$ of relative dimension 2 ($k$ is a field of positive characteristic). They then consider the family of genus 2 stable curves over $\mathbb{P}_k^1$ whose Jacobian is $A$.

For a ppa surface over a field, one can easily recover the stable curve it comes from as follows. Let $L$ be the ample line bundle corresponding to the polarization; it is known that $H^0(A,L)$ is one-dimensional, so all global sections share a common zero set which is a curve $C$, and I believe it is not too hard to show that $\mathrm{Pic}^0(C) \simeq A$. I think uniqueness of the zero set makes this construction functorial: there really is only one theta divisor.

I struggle to make this construction work in families. If $L$ denotes a relatively ample sheaf representing the polarization, I want to argue that it has a global section (this seems doable), whose zero set is flat over the base so as to define a relative Cartier divisor (this is the challenging part), which would be a family of genus 2 stable curves whose Jacobian recovers the abelian scheme. Making this construction functorial requires some uniqueness; but the fiberwise uniqueness of the theta divisor does not clearly imply global uniqueness, especially over a non-reduced base.

As a final remark, and maybe an alternative route, de Jong and Oort's construction comes from Moret-Bailly's Familles de courbes et de variétés abéliennes sur $\mathbb{P}^1$, II Exemples, in which he recovers the theta divisor as follows. Let $L$ once again be a line bundle representing the principal polarization. He then claims that the theta divisor is the support of the cokernel of the adjunction morphism $$f^* f_* L \longrightarrow L.$$ Let $Z$ denote the support, and $D$ the theta divisor; I can see why $Z \subseteq D$ (the morphism is an iso outside of $D$), but the reverse inclusion is unclear. It is also not so clear to me that this construction could be functorial.

Any help is greatly appreciated !