The real work is in the case of bad reduction, as otherwise one can use finite etale torsion levels of the entire elliptic curve (the crux being that passage to the special fiber is an equivalence of categories between finite etale schemes over a henselian local ring and over its residue field, so in particular reduction is a faithful functor on such schemes). Below we give a method that treats all cases of semistable reduction in all dimensions by a uniform method, in fact ultimately using the same "crux" mentioned above (with help from Zariski's Main Theorem to get around the headache of quasi-finite maps that may not be finite).
Any specific endomorphism is defined over an actual number field $K$ inside $\overline{\mathbf{Q}}$, and in case of semistable reduction the formation of the identity component of the special fiber of the Neron model commutes with any local extension of the discrete valuation ring. Thus, we can work relative to the discrete valuation ring $O_{K, \mathfrak{p}}$ (localization at a prime over $p$). It is sufficient to prove more generally that if $A$ and $B$ are abelian varieties (or just elliptic curves) over the fraction field $F$ of a discrete valuation ring $R$ with residue field $k$ and their Neron models $\mathcal{A}$ and $\mathcal{B}$ over $R$ have $k$-fibers with semistable reduction then the natural map $${\rm{Hom}}_F(A,B) = {\rm{Hom}}_R(\mathcal{A}, \mathcal{B}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k, \mathcal{B}^0_k)$$ is injective.
We may and do apply scalar extension on $R$ so that it is complete (or just henselian is good enough for what follows, if one wants to be more "algebraic"). We may also assume $A \ne 0$, and that $A$ is $F$-simple (since in the case of semistable reduction isogenies between abelian varieties induce isogenies between identity components of special fibers of Neron models, as we see by factoring an isogeny through multiplication by some nonzero integer).
Pick a prime $\ell \ne {\rm{char}}(k)$ and consider the $\ell$-power torsion $\mathcal{A}^0[\ell^n]$, where $\mathcal{A}^0$ is the open "relative identity component" (i.e., the open complement in $\mathcal{A}$ of the closed locus of non-identity components in the closed fiber). The hypothesis of semistable reduction implies (by fibral considerations, due to flatness of the Neron model over $R$) that these are quasi-finite etale separated commutative $R$-groups with special fiber $\mathcal{A}^0_k[\ell^n]$ that has order at least $\ell^n$ (recall we arranged $A \ne 0$).
By Zariski's Main Theorem, any quasi-finite etale separated scheme $X$ over a henselian local ring $C$ is uniquely a disjoint union $X_{\rm{f}} \coprod X'$ of a finite $C$-scheme $X_{\rm{f}}$ and a $C$-scheme $X'$ with empty special fiber. The formation of $X_{\rm{f}}$ is functorial in $X$ and compatible with direct products, so if $X$ is a $C$-group then $X_{\rm{f}}$ is an open and closed $R$-subgroup.
We conclude that $\mathcal{A}^0[\ell^n]_{\rm{f}}$ is a finite etale $R$-group with order at least $\ell^n$. For finite etale schemes over a henselian local ring $C$, passage to the special fiber sets up an equivalence of categories (a very familiar fact for $C$ a complete discrete valuation ring with perfect residue field, where it amounts to the Galois theory of the residue field encoding the structure of unramified extensions of the fraction field). Thus, the natural map $${\rm{Hom}}_R(\mathcal{A}^0[\ell^n]_{\rm{f}}, \mathcal{B}^0[\ell^n]_{\rm{f}}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k[\ell^n],\mathcal{B}^0_k[\ell^n])$$ is an equality! In particular, if $h:A \rightarrow B$ is an $F$-homomorphism whose induced map $\mathcal{A}_k^0 \rightarrow \mathcal{B}_k^0$ vanishes then the map between Neron models vanishes on $\mathcal{A}^0[\ell^n]_{\rm{f}}$, and hence $h$ vanishes on the generic fiber $(\mathcal{A}^0[\ell^n]_{\rm{f}})_F$ for all $n \ge 1$. This generic fiber has order at least $\ell^n$, so $\ker h$ contains finite etale $F$-subgroups of order at least $\ell^n$ for all $n \ge 1$.
To conclude that $\ker h = A$, it now suffices to show that if $A$ is an $F$-simple abelian variety over any field $F$ and $H$ is a closed $F$-subgroup scheme of $A$ containing finite etale $F$-subgroups $H_n$ with unbounded order as $n$ varies then $H=A$. Consider the subgroup of $H(F_s)$ generated by the groups $H_n(F_s)$. This is a ${\rm{Gal}}(F_s/F)$-stable subgroup of $H(F_s)$, so the Zariski closure in $H_{F_s}$ of that subgroup is a closed $F_s$-subgroup scheme $Z'$ of $H_{F_s}$ that is preserved by the natural $F_s/F$-descent datum on $H_{F_s}$ encoding its $F$-descent $H$ and is geometrically reduced over $F_s$, hence is $F_s$-smooth. Thus, by Galois descent $Z'$ descends to an $F$-smooth closed $F$-subgroup $Z \subset H$ that contains every $H_n$ by design. For the abelian variety $Z^0 \subset A$, we have that $Z/Z^0$ is finite, so the orders of the $F$-subgroups $Z^0 \cap H_n$ are unbounded as $n$ varies. This implies $Z^0 \ne 0$. But $A$ is $F$-simple by hypothesis (!), so $Z^0 = A$. By design we have $Z^0 \subset H$, so $H=A$ as desired.