Have you tried looking at Sinnott's paper where he re-proves Ferrero-Washington for $\mathbb{Q}$? It is in Invent. Math, 1984 vol. 75 (2) pp. 273-282.

He proves that to compute the $\mu$-invariant of a function that can be expressed as $\Gamma$-transform of a power series, it is enough to know the $\mu$-invariant of the series. He then applies this to the construction of the $p$-adic $L$-function of Iwasawa where an *explicit* expression (page 282 and equation (4.3) on page 280) of the paper can be found. Since this ''explicit expression'' comes from the Euler system of cyclotomic units and we now dispose of the Euler system of elliptic units (*i.e.* we now call it in such a way) for the cyclotomic extension of your imaginary quadratic field, it is plausible that Sinnot's argument applies. But if Coates and Rubin have doubts there must be something tricky behind.