I'll post (in this thread) the full solution when I get there

It turns out to be fairly straightforward.
Setting $ENV{GIO_MODULE_DIR} or $ENV{GIO_EXTRA_MODULES} to the location of libgiognutls.dll is all that's needed to make that dll locatable, and it doesn't seem to matter if one changes the name of that dll.
Apart from that, I just need to ensure that all of the dlls that libgiognutls.dll needs to pull in can be found. This can be achieved by (eg) making sure that they're all in $ENV{PATH}.

Cheers,
Rob