Let $(\mathcal{M},g)$ be a four-dimensional Lorentzian manifold and $\Sigma$ a Riemannian hypersurface such that $\mathcal{M}=\mathbb{R}\times\Sigma$ with metric of the form $g=-\alpha^{2}d t^{2}+h_{t}$ where $\alpha$ is some positive function, $t:\mathcal{M}\to\mathbb{R}$ is a smooth function and $h_{t}$ a family of Riemannian metrics on $\Sigma$.
Let $A\in\Omega^{1}(\mathcal{M})$ be a $1$-form. Clearly, I can write $$A=A_{0}dt+A_{1}$$
where $A_{0}:=A(\partial_{t})\in C^{\infty}(\mathcal{M})$ and where $A_{1}:=A-A_{0}$ is a $t$-dependent $1$-form on $\Sigma$. Now, this essentially gives the isomorphism
$$\Omega^{1}(\mathcal{M})\cong C^{\infty}(\mathcal{M})\oplus\Gamma(\mathrm{pr}^{\ast}T^{\ast}\Sigma)$$
where $\mathrm{pr}:\mathcal{M}\to\Sigma$ denotes the projection. (This also respects the ismorphism $T^{\ast}\mathcal{M}\cong \underline{\mathbb{R}}\oplus\mathrm{pr}^{\ast}T^{\ast}\Sigma$ obtained in this answer).
Question: How is the information regarding the transformation law of tensors encoded in the space $C^{\infty}(\mathcal{M})\oplus\Gamma(\mathrm{pr}^{\ast}T^{\ast}\Sigma)$?
For example, if $A_{0}\in C^{\infty}(\mathcal{M})$, then also $\partial_{t}A_{0}\in C^{\infty}(\mathcal{M})$. Similarly, $\partial_{t}A_{1}\in \Gamma(\mathrm{pr}^{\ast}T^{\ast}\Sigma)$ for $A_{1}\in \Gamma(\mathrm{pr}^{\ast}T^{\ast}\Sigma)$ (since the time-derivative of a time-dependent 1-form is again a time-dependent 1-form on $\Sigma$). However, in general $\partial_{t}A\notin\Omega^{1}(\mathcal{M})$ since $\nabla_{\partial_{t}}A\neq \partial_{t}A$. Where is my thinking error?