Are there any irreducible representations of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ over $\mathbb Q_l$ that are unramified away from $l$ and crystalline at $l$ but are not known to arise from the etale cohomology of smooth and proper stacks over $\operatorname{Spec}\mathbb Z$?

Thee simplest examples of unramified crystalline Galois representations known, other than powers of the cyclotimc character, are the two-dimensional Galois representations arising from level $1$, weight $n+1$ modular forms. These live inside the etale cohomology of the stack $\overline{M}_{1,n}$, which is indeed smooth and proper. More examples come from other $\overline{M}_{g,n}$, but these are the only examples I am aware of!

The situation for schemes is very different where, as far as I can tell, no Galois representation other than powers of the cyclotomic character is known to arise from smooth and proper schemes over $\mathbb Z$.