I did experience this same issue with the project I work on. You can overcome it by explicitly loading the modulemaps file of the Swift package's target through the OTHER_SWIFT_FLAGS build setting. Assuming you have a .xcconfig file (which's great in this use case, as you can add a comment to explain this weird logic), you can add this:
// SPM package modulemaps are not automatically available to ObjC header files while they are being parsed for the bridging
// header. In most cases this shouldn't be needed, because ObjC headers generally should forward-declare types rather than
// importing other headers. Until Swift 5.9, however, forward declared types are not imported to Swift.
//
// https://github.com/apple/swift-evolution/blob/main/proposals/0384-importing-forward-declared-objc-interfaces-and-protocols.md
//
// This overlaps several discussions of various relevancy in the Apple-sphere, but all of which help shed some light on the
// underlying issues. This is not strictly a bug. It is a mix of some missing features and "that's just how things work."
//
// https://bugs.swift.org/browse/SR-15154
// https://github.com/apple/swift-package-manager/issues/4531
// https://developer.apple.com/forums/thread/650935
// https://forums.swift.org/t/obj-c-package-with-swift-dependency-not-accessible-in-swift/63137
// https://developer.apple.com/forums/thread/120152
// https://pfandrade.me/blog/mixing-swift-objective-c-spm-and-static-frameworks/
//
OTHER_SWIFT_FLAGS = -Xcc "-fmodule-map-file=${OBJROOT}/GeneratedModuleMaps-${PLATFORM_NAME}/AnimalSwift.modulemap"
The root cause of the issue is thoroughly explained by my awesome colleague in the comment above. I did try it with your sample project and it worked. Once you load the modulemaps file, you can then uncomment your @import in the header file which will work.