A small clarification in case anyone gets confused in the future ... I think there is a typo in the question. The question says that if $p$ is a prime of good ordinary reduction for an elliptic curve $E$, we have an exact sequence as $\text{Gal}(\overline{K}/K)$-modules $$0 \to \mathbf{Z}_p(-1) \to T_pE \to \mathbf{Z}_p \to 0.$$ However, this is not an exact sequence of $\text{Gal}(\overline{K}/K)$-modules. It is an exact sequence of $\text{Gal}(\overline{K}/\hat{K^{ur}})$-modules, where $\hat{K^{ur}}$ is the (completion of) the maximal unramified extension of $K$.

So what does $T_pE$ look like as a $\text{Gal}(\overline{K}/K)$-module? Here, unfortunately, we can say far less. Assume that $p$ is a prime of good ordinary reduction for $E$. Then all we know is that there is an exact sequence of $\text{Gal}(\overline{K}/K)$-modules $$0 \to T^{'} \to T_pE \to T^{''} \to 0$$ where $T^{'}$ and $T^{''}(-1)$ are unramified as $\text{Gal}(\overline{K}/K)$-modules*. I don't think we can say what $T^{'}$ and $T^{''}$ look like exactly as $\text{Gal}(\overline{K}/K)$-modules because that would depend on the specific curve $E$.

*This fact from Kato's Asterisque 295 paper "p-adic hodge theory and zeta values of modular forms" Proposition 17.1, part (iii). Set $T^{'}$ and $T^{''}$ as in Proposition 17.2.

This is not an answer, but a clarification about a typo in the question. The question says that if $p$ is a prime of good ordinary reduction for an elliptic curve $E$, then we have an exact sequence of $\text{Gal}(\overline{K}/K)$-modules $$0 \to \mathbf{Z}_p(-1) \to T_pE \to \mathbf{Z}_p \to 0.$$ However, the above exact sequence is not an exact sequence of $\text{Gal}(\overline{K}/K)$-modules. It is in fact an exact sequence of $\text{Gal}(\overline{K}/\hat{K^{ur}})$-modules, where $\hat{K^{ur}}$ is the (completion of) the maximal unramified extension of $K$.

So what does $T_pE$ look like as a $\text{Gal}(\overline{K}/K)$-module? Here, unfortunately, we can say far less. Assume that $p$ is a prime of good ordinary reduction for $E$. Then all we know is that there is an exact sequence of $\text{Gal}(\overline{K}/K)$-modules $$0 \to T^{'} \to T_pE \to T^{''} \to 0$$ where $T^{'}$ and $T^{''}(-1)$ are unramified as $\text{Gal}(\overline{K}/K)$-modules*. I don't think we can say what $T^{'}$ and $T^{''}$ look like explicitly as $\text{Gal}(\overline{K}/K)$-modules because that would depend on the specific curve $E$.

*This fact from Kato's Asterisque 295 paper "$p$-adic hodge theory and zeta values of modular forms", Proposition 17.1, part (iii). Set $T^{'}$ and $T^{''}$ as in Proposition 17.2.