There are two issues. One I understand and one I don't. Let $H=H^1_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V_{\mathbb{Z}_p}(f)$ and $f$ is the modular form associated to the isogeny class of $E$.
(1) What is $T$ ?
I thought $T$ is the Tate-module of the $X_1$-optimal curve, but I was told this is wrong. I truely hope someone will answer this sub-question. Maybe Tony Scholl could say something about it.
(2) Are Kato's elements integral in $H$ ?
There are two kinds of them. The $z_{\gamma}$ and the ${c,d}$ $z_{m}$. (sorry I don't seem to be able to produce indices before the symbol inMathJax) The latter are in $H$, see 8.1 of Kato, but they depend on the choices of $c$ and $d$. They are useful for bounding the Selmer group as, for a fixed $c$ and $d$ they form an Euler system.
The $z_{\gamma}$ instead is linked to the $p$-adic $L$-function and they are independent of the choices. They are obtained by dividing by $\mu(c,d)$, page 229 of Kato. So they need not be integral anymore. The appendix A in Delbourgo's book "Elliptic curves and big Galois representations" discusses this in detail. The fact that $z_\gamma$ is not known to be integral is linked to the integrality of the $p$-adic $L$-function (the vanishing of the $\mu$-invariant is an even harder question, I believe). But even the integrality of the $p$-adic $L$-function (known for the $X_1$-optimal curve by Greenberg-Vatsal) does not seem to imply that $z_\gamma$ is integral in $H$, maybe it does in $H^1_{\mathrm{Iw}}(\mathbb{Q}_p,T)$.
To prove that $z_{\gamma}$ is integral, one uses that $H$ is a free $\Lambda$-module of rank 1, e.g. as shown in 12.4.(3) if $T/pT$ is irreducible. In fact it is not hard to show that $H$ is free also if $E(\mathbb{Q})[p]$ is trivial. For instance, we really only have to worry when $T/pT$ is irreducible rather than just whether the representation is surjective.
If $T$ were known to be $T_pE$ as in (1), then my corrections aimed to show that one can still deduce the divisibility, because the denominators introduced by $\mu(c,d)$ are not too big, i.e. $Z(f,T)$ has finite index in $Z$, see 13.12 in Kato. This would also prove that the $p$-adic $L$-function is integral and that $\mu\geq 0$, but it would not say anything about $\mu=0$.
Now, in the proof of the divisibility in the main conjecture as in 12.5.(3), one also needs to apply the Euler system method. And here we need the element $\binom{1\ 1}{0\ 1}$ in the image of the Galois representation, see Hyp($K_{\infty},T)$ in Rubin's book. So the integrality won't help, yet. My flawed paper was based on the fact that if $E[p]$ is reducible, then we can circumvent the problem in the Euler system argument, by knowing that the $\mu$-invariant for class groups vanish.
In summary, I am certain everything is integral if $T/pT$ is irreducible and I could imagine that one can prove it too in the reducible case, maybe assuming Steven's conjecture on $\mu=0$. For Kato's divisibility on the other hand, the surjectivity of the representation to $\mathrm{GL}_(\mathbb{Z}_p)$ is still needed.