Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there be between the local Tamagawa number $c_v(A) = [A(K_v): A_0(K_v)]$ and the cardinality of the $p$-primary subgroup $A(K_v)(p)$ of $A(K_v)$?

I am aware of the recent work of Lorenzini, which considers possible cancellations of the ratio \begin{align*} \frac{\prod_{v \subset \mathcal{O}_K} c_v(A)}{\vert A(K)_{\operatorname{tors}}\vert}, \end{align*} and shows for instance this ratio for $A$ an elliptic curve defined over $K ={\bf{Q}}$ is always greater than or equal to $\frac{1}{5}$.

The question comes up naturally in a certain Euler characteristic computations (e.g. for the
$p^{\infty}$-Selmer group of $A$ defined over some profinite Galois extension $K_{\infty}$ of $K$, where certain primes of $K$ are known to split completely, and hence where the torsion subgroup $A(K_{\infty})_{\operatorname{tors}}$ is known to be finite). In particular, granted the refined conjecture of Birch and Swinnerton-Dyer, it is apparent from these computations that $c_v(A)$ for a prime of bad (multiplicative) reduction $v$ is given essentially by the quotient `\begin{align*}\frac{\vert H^1(G_w, A(K_{\infty, w}))(p)\vert }{\vert H^2(G_w, A(K_{\infty, w}))(p)\vert }. \end{align*}`

Here, $K_{\infty, w}$ is the union of all completions at primes above $v$ in $K_{\infty}$, and $G_w$ denotes the Galois group $\operatorname{Gal}(K_{\infty, w}/K_v)$.
If $A$ has good ordinary reduction at all primes above $p$ in $K$, then it is possible to use the Coates-Greenberg theory of deeply ramified extensions to characterize the local factor at $v$ in the Euler characteristic formula coming from this quotient as $\vert \widetilde{A}(\kappa_v)(p)\vert^2$, where $\widetilde{A}$ is the reduction of $A$ mod $v$, and $\kappa_v$ the residue field at $v$ (which is consistent with B+S-D). But, this characterization seems to break down when $A$ does not have good ordinary reduction at $v$, and I have not found any calculations in the literature for this case of bad reduction. Hence why I ask about any possible known relation to torsion. Sorry if the question is silly!