I kept trying throughout the day on the 8th without any luck. Today I've been trying once again but with increased frequency, hitting the refresh button over and over more times than I can count. At about 10:20, to great surprise, it loaded a single date in early October. In the minute it took me to check calendars that I could make that day, when I clicked to confirm I got the no available dates message again. I was gutted.
But around 2 hours later I hit the lottery again and was offered a single date in September, which I confirmed immediately without even checking my schedule - I'll make it work!
Best of luck to any others trying, it seems to take a lot.