Closure of Star-Free Languages under Substitution
I do not understand your proof sketch. Where do you want to subtitute $\sigma(1)$ under a complement. The only thing substituable in the star-free expression for $1^*$ is $0$. But 0, or $\sigma(0)$ is an independent element in the substitution, it is in no way related to the complement of $\sigma(1)$. So this can in no way make a proof.

Actually, if there was such a closure property, it should be easy to find in
the literature on star-free RE. But all closure results under substitution put
constraints on the type of substitution, such as being length
preserving, or being injective.
See in particular papers by Jean-Eric Pin and others.

Also,Pin defines star-free substitutions on $\Sigma$ as requiring an additional property, namely that $(\sigma(\Sigma))^*$ is star-free. (though I did not check
the precise role of this property)

It seems that you have just proved that star-free languages are not closed under star-free substitution as you define it.

This does not seem too surprising, since non injective substitution does not in general commute with complement or distribute with intersection.

Among other papers:
Jean-Eric Pin, H. Straubling, and D. Thérien. Some results on the generalized star-height problem. May 12 2002.