I can simplify the workaround steps for everyone a bit.
Here's a link to a modified copy of adapter_fanfictionnet.py. Just open your FanFictionDownLoader.zip file as described above and replace the existing one with this one and when you restart Calibre it should work; I tested it locally and it works for me. I only added the one extra line to modify FFDL's user agent.
Thanks, shesgottaread, for digging up this fix in the first place (and JimmXinu for putting up with us flailing around in this thread like this
. I'm sure this will get sorted out soon, in the meantime this can tide over anyone with a more urgent need for a little light reading.