Update: I feel a bit goofy. After posting, I continued attempting to resolve the problem, and I appear to have found a resolution only a few minutes late.
Removing typechecking from the swiftc call yields a header file. That is, I now call swiftc:
swiftc Sources/[PackageName]/Foo.swift -module-name [PackageName] -cxx-interoperability-mode=default -emit-clang-header-path [HeaderName].h
... And the header is generated. I hope this post is useful to someone, regardless.