I'm not aware of any context where we have a fixed map $\pi$ but consider different structures on the domain of $\pi$ with respect to which $\pi$ remains a homomorphism, and this is the only sort of context I can think of where "$A$ is the domain of $\pi$" would seem inappropriate to me. So I don't think such a term exists.
(Contrast this with "$\pi$ is a homomorphism from $A$" - this language is useful since we do often consider multiple homomorphisms out of the same structure.)
EDIT: you clarify that this is in fact the context you're in. I stand by my claim that this context is rare enough that there probably isn't a standard term in the literature. That said, the term which fits the bill most cleanly for me is "compatible," and I would suggest the phrase "$A$ is a compatible domain for $\pi$," or - if you only ever talk about varying domains and not codomains - "$A$ is compatible with $\pi$." But of course unless I'm wrong and there is after all a standard term this should be defined explicitly in your text.
Alternatively, a phrase like "$A$ is a domain object for $\pi$" has a categorical flavor which may be desired (this is motivated by Ty Jensen's comment above). Personally I think this sort of thing is clunkier, but I could see it being preferred.