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, [this][1] post.


**NB:** If anyone has anything interesting to add, I would be more than happy to hear it/accept it as an answer!


  [1]: http://mathoverflow.net/questions/53014/in-which-ways-can-the-isogeny-theorem-fail-for-local-fields