So, the solution to the whole problem ended up being a lot simpler
than I had feared.
Install kernel-devel rpm's from the Fedora DVD.
Sneakernet the e1000 tarball from Intel's website to the destination machine.
# tar zxvf e1000-abc.tar.gz
# cd e1000-abc/src
# make install
# chmod 744 /lib/modules/kernel-xyz/drivers/net/e1000/e1000.ko
The key step, which I didn't know to do the first time, was install
kernel-devel. I thought I needed to install the kernel-source, and
build the kernel to generate version.h.
So, easily solved once you know the solution.