It took me a while to realise that this is an interesting question. The formulation above makes it sound like a computational problem for a specific curve, so let me first reformulate it:

Let $E/k$ be an elliptic curve over a number field $k$ and let $\varphi:E \to E'$ be a cyclic isogeny of degree $p^2$ for some prime $p$. To determine how the $\mu$-invariant (with respect to some $\mathbb{Z}_p$-extension $k_{\infty}$, say the cyclotomic one) of $E$ changes one has to look at how the complex conjugations act on the kernel $C$ of $\varphi$. So if it changes by $\mu_E =\mu_{E'} + 2$, how does the $\Lambda$-module change? By two additional $\Lambda/p$ or by one additional $\Lambda/p^2$ or otherwise?

The analytic side of the main conjecture would not help us with that. On the algebraic side, we get a exact sequence of $\Lambda$-modules $$ \to H^2(C)^* \to X(E') \to X(E) \to H^1(C)^* \to $$ where $X(E)$ and $X(E')$ are the Pontryagin duals of the Selmer groups of $E$ and $E'$ over $k_{\infty}$ and $H^i(C)^*$ are the Pontryagin duals of the "Selmer groups" of $C$; i.e., $H^i(C)$ is the kernel of global-to-local map on the $H^i(k_{\infty}, C)$. So the question is what is the structure of the $\Lambda$-modules $H^i(C)^*$.

In the particular cases, asked above, $k=\mathbb{Q}$ and $p=3$, and the kernel $C$ is in a short exact sequence $$0\to \mu_p \to C \to \mu_p \to 0$$ of Galois-modules, but it is not $\mu_{p^2}$. It is possible to see that $H^1(\mu_p)^* $ is pseudo-isomorphic to $\Lambda/p$ and $H^2(\mu_p)^*\sim 0$. From this we find that $H^1(C)$ sits between two $H^1(\mu_p)$, but also that $H^1(C)[p] \sim H^1(\mu_p)$. Hence $H^1(C)^* \sim \Lambda/{p^2}$. The two examples have moreover $X(E') \sim 0$, so $X(E)\sim \Lambda / p^2$.