The formal affine line is an etale stack!

I am tasked to prove the formal affine line $\hat{\mathbb{G}}_a$, seen as the functor from animated rings to set $$Ani(Ring)\to Ani$$ $$R\mapsto Nil(\pi_0(R))$$ taking an animated ring to the nilradical of its underlying static ring, to be an etale stack i.e. I have to show that it satisfies etale descent. If we spell this out this means that for $A\to B$ an etale cover, the natural map $$\hat{\mathbb{G}}_a(A)\to lim(\hat{\mathbb{G}}_a(B)\to \hat{\mathbb{G}}_a(B\otimes_A B)\to \cdots)$$ is an equivalence. I was told that this is in fact not an fpqc sheaf, any clarification will be much appreciated, also do not hesitate to ask for clarifications. My attempt: since we can compute limit of sets in Ani naively, we can truncate the totalization, and reduce to showing $$Nil(\pi_0(A))\to eq(Nil(\pi_0(B))\to Nil(\pi_0(B)\otimes_{\pi_0(A)}\pi_0(B)))$$ being an equivalence. But the map $\pi_0(A)\to \pi_0(B)$ is faithfully flat and hence in particular injective, for surjectivity: let $x\in eq(...)$ then this is an element x in B such that $x^n = 0$ and $1\otimes x = x\otimes 1$, in particular there exists $a\in A $ going to x by standard faithfully flat descent, but by injectivity of the map also $a\in A$ must be nilpotent. My problem is that this seems to suggest $\hat{\mathbb{G}}_a$ to be an fpqc stack! but my professor told me it is false.