Hi Heinrich!

I don't know. But you could take a look at Carlos Simpson's general definition of "filtered object" on pages 24/25 of his paper The Hodge filtration on non-abelian cohomology: Roughly, a filtered X is a $\mathbb{G}_m$-equivariant map from an X to $\mathbb{A}^1$, so you could get your moduli space as a mapping space, or as an object of the "comma site" of maps of objects of your site into $\mathbb{A}^1$...

I don't know. But you could take a look at Carlos Simpson's general definition of "filtered object" on pages 24/25 of his paper The Hodge filtration on non-abelian cohomology: Roughly, a filtered X is a $\mathbb{G}_m$-equivariant map from an X to $\mathbb{A}^1$, so you could get your moduli space as a mapping space, or as an object of the "comma site" of maps of objects of your site into $\mathbb{A}^1$...