The original question, as stated, has a negative answer. Namely, it is not true that the induced map
$$\text{Hom}(A,A')\otimes\mathbb{Z}_p\to \text{Hom}_{F,V,\text{Fil}}(H^1_{\text{crys}}(\mathscr{A}'_k/W(k)),H^1_\text{crys}(\mathscr{A}_k/W(k)))\qquad (1)$$
is an isomorphism.
For simplicity let $K=\mathbb{Q}_p$ (so $k=\mathbb{F}_p$). By the fully faithfulness of the $D_\text{crys}$ functor we know that
$$\text{Hom}(H^1_{\acute{e}\text{t}}(A',\mathbb{Q}_p),H^1_{\acute{e}\text{t}}(A,\mathbb{Q}_p))\to \text{Hom}(D_\text{crys}(H^1_{\acute{e}\text{t}}(A',\mathbb{Q}_p)),D_{\text{crys}}(H^1_{\acute{e}\text{t}}(A,\mathbb{Q}_p)))$$
is an isomorphism but
$$D_\text{crys}(H^1_{\acute{e}\text{t}}(A',\mathbb{Q}_p))=H^1_{\text{crys}}(\mathscr{A}'_k/K)$$
and similarly for $A$. Thus, if $(1)$ were true then the map
$$\text{Hom}(A,A')\otimes\mathbb{Z}_p\to \text{Hom}(H^1_{\acute{e}\text{t}}(A',\mathbb{Q}_p),H^1_{\acute{e}\text{t}}(A,\mathbb{Q}_p))$$
and thus the map
$$\text{Hom}(A,A')\otimes\mathbb{Z}_p\to\text{Hom}(T_p A,T_p A')$$
is a isomorphism. But, this is Tate's isogeny conjecture for $\mathbb{Q}_p$ which is false. See, for example, thisthis post.
NB: If anyone has anything interesting to add, I would be more than happy to hear it/accept it as an answer!
