Is that what you have ? (I get that SHA1 digest for the distro that I just downloaded - so that part looks fine to me.) If you get a different SHA1 digest for the distro you downloaded then, either you've grabbed a different distro, or you've got a corrupted distro.
SHA1 was correct.
When I checked where objdump was, however, I found it in two places: my separate MSYS installation, and the strawberry one. Once I removed all of the MINGW stuff from the path, strawberry no longer had any trouble compiling XS stuff. Maybe I had a wrong version of MINGW or something; either way, I'm happy it works now. Thanks!
It's probably a 32-bit version of MinGW - which will still work fine on your 64-bit system, btw. But there are incompatibilities between the 32-bit binaries it builds/uses and the 64-bit binaries that Strawberry builds/uses - and it's one or more of those incompatibilities that has bitten you in this instance.
I'm pleased it's finally working. Thank you for the feedback.