Dear David,
There's also a less direct way to see that the category of affine schemes is not Mal'cev, but one that is more in line with your motivation -- namely, by exhibiting an internal category which is not a groupoid: for instance, take the multiplicative monoid. As a functor from commutative rings to monoids, it just forgets addition, and is co-represented by the ring $\mathbb{Z}[t]$, with co-multiplication sending $t$ to $xy\in\mathbb{Z}[x,y]$ and the co-unit given by evaluating at $t=1$.