And discovered that, despite what I'm being told by LJ support, it's definitely blocking me. I'm getting a "403" back - which means "forbidden".
So I've updated my support request to let them know that, along with the IP address that it's making the request from (which was a bit of fun to find out in the first place - as I had to make a request to an external location and see what IP it saw the request as coming from - it's 184.108.40.206).
Now, I guess, I wait!
Original post on Dreamwidth - there are comments there.