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If you look at Theorem 1 in Hida's paper quoted in my comment, I think that you'll get what you are looking for.

Since $F=\mathbb{Q}$ (I stick to Hida's notation) and you work with the maximal unramified-outside-of-$\mathfrak{p}$-extension, the prime-to -$p$ part of the conductor to be considered in condition (S) is trivial and $p$ is certainly split in $F$. So the theorem applies. Now, split the maximal unramified-outside-of-$\mathfrak{p}$ extension $K(\mathfrak{p}^\infty)/K$ as the compositum $K(\mathfrak{p}^\infty)=K'K_\infty$, where $[K':K]$ is prime to $p$. The theorem tells you - if you are willing to believe it - that the projection of the $p$-adic $L$-function $\varphi$ (seen as a measure on the big Galois group $\mathrm{Gal}(K(\mathfrak{p}^\infty)/K)=\Gamma\times\mathrm{Gal}(K'/K)$ where $\Gamma$ is the Galois group of your extension) to a measure on $\Gamma$ has trivial $\mu$ invariant: this projection, indeed, corresponds to the branch character $\psi_0=\mathrm{id}$.

A last word of warning: some techniques resorting from the study of Hilbert-Blumenthal Abelian Varieties require that the totally real base field be different from the rationals. I went quickly down through rapidly skim the paper, without seeing any sort of hypothesis $F\neq\mathbb{Q}$, but if you intend to apply it in a research paper, I'd advise you to double-check this assumption.

1

If you look at Theorem 1 in Hida's paper quoted in my comment, I think that you'll get what you are looking for.

Since $F=\mathbb{Q}$ (I stick to Hida's notation) and you work with the maximal unramified-outside-of-$\mathfrak{p}$-extension, the prime-to -$p$ part of the conductor to be considered in condition (S) is trivial and $p$ is certainly split in $F$. So the theorem applies. Now, split the maximal unramified-outside-of-$\mathfrak{p}$ extension $K(\mathfrak{p}^\infty)/K$ as the compositum $K(\mathfrak{p}^\infty)=K'K_\infty$, where $[K':K]$ is prime to $p$. The theorem tells you - if you are willing to believe it - that the projection of the $p$-adic $L$-function $\varphi$ (seen as a measure on the big Galois group $\mathrm{Gal}(K(\mathfrak{p}^\infty)/K)=\Gamma\times\mathrm{Gal}(K'/K)$ where $\Gamma$ is the Galois group of your extension) to a measure on $\Gamma$ has trivial $\mu$ invariant: this projection, indeed, corresponds to the branch character $\psi_0=\mathrm{id}$.

A last word of warning: some techniques resorting from the study of Hilbert-Blumenthal Abelian Varieties require that the totally real base field be different from the rationals. I went quickly down through the paper, without seeing any sort of hypothesis $F\neq\mathbb{Q}$, but if you intend to apply it in a research paper, I'd advise you to double-check this assumption.