Denote by $\Phi$ the quotient of $\mathcal A^\vee$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $A^\vee$, by the connected component of $0$ of $\mathcal A^\vee$. Then it is indeed true that the *Tamagawa number* $c_p$ as defined in *$L$-functions and Tamagawa number of motives* (Bloch, Kato) or in *Autour des conjectures de Bloch-Kato* (Fontaine, Perrin-Riou) is then equal up to a $p$-adic unit to $|\Phi[p^\infty]|$, but this is not quite obvious.

The main step of the proofs are as follows:

- First, relate $\Phi$ to the formal group of $A^\vee$, and then through the $\log$ map to $\operatorname{Lie}_{\mathbb Z_p}A^{\vee}$ (which is also the tangent space of the formal group with values in $\mathbb Z_p$).
- Then notice that this induces isomorphisms between the determinant of $\operatorname{Lie}_{\mathbb Z_p}A^{\vee}$ and the determinant of the first Bloch-Kato cohomology group $H^1_f(\mathbb Q_p,T_pA)$ of the $p$-adic Tate module of $A$. This isomorphism involves finite corrective terms which come from the fact that the formal group is a subgroup with finite index of $H^1_f(\mathbb Q_p,T_pA)$ (and similarly for the image of the $\log$ with respect to $\operatorname{Lie}_{\mathbb Z_p}A^{\vee}$).
- Now the Tamagawa number $c_p$ arises as an alternate product of these corrective terms and of the Euler factor at $p$ (because the exponential map of Bloch-Kato coincide in this case with $\log$).
- Finally, relate this alternate product to $\Phi$. This involves the classical description of $\Phi$ in terms of the Neron model as well as a study of the reduction of $A^{\vee}$.

Personally, I don't find any of these steps particularly easy, and the last one certainly isn't. I don't have a good reference to mention that would cover all of them. A good but hard reference for the hardest part is SGA7, Exposé IX.

One thing that I would mention is that, as you say, the computations of the proof above involve in several steps the choice of a $\mathbb Z_p$-basis, but as this plays a role both in the normalization of maps with target $\operatorname{Lie}A^\vee$ as well as in the definition of $c_p$, the resulting computation does not depend on the choice of the basis (up to a $p$-adic unit, which plays no role in the Tamagawa number conjectures of Bloch-Kato).