I am a bit confused and wondering where a contradiction is in the following argument. Let's do everything over $\mathbb{C}$.

Let $X\to C$ be an elliptic surface over $\mathbb{P}^1$ with no multiple fibers.

The following facts are well known (c.f. Friedman Algebraic Surfaces and Algebraic Vector Bundles)

1. The canonical bundle is given by the formula $$K_X = \pi^*(K_C \otimes L)$$ where $L$ is a line bundle of degree $d \ge 0$ in particular it satisfies $K_X^2 = 0$.
2. Noether's formula implies $$e(X) = 12 \chi(\mathcal O_X) = 12d$$ where $e(X) = c_2(X)$ is the topologial Euler characteristic of $X$.

The Euler characteristic can be computed by understanding the singular fibers using cut and paste since the smooth fibers $F$ have $e(F) = 0$.

On the other hand, it is known (e.g. here) that there are isotrivial ellitpic surfaces over $\mathbb{A}^1 $ with a single singular fiber $F_0$ of type $I_0^*, II, II^*, III, III^*, IV, IV^*$ with $e(F_0) \in \{ 1,2,3,5,7,8,9\}$.

Now construct $X$ by gluing one of these surfaces over $\mathbb{A}^1$ to a trivial smooth elliptic fibration over $\mathbb{A}^1$ along $\mathbb{A}^1 - \{ 0\}$. We get a isotrivial surface over $\mathbb{P}^1$ with $e(X) \in \{ 1,2,3,5,7,8,9\}$ which is not a multiple of 12.

Where is the error?