I would call again. When I phoned yesterday there was availability and it was only 90+6. Today, when I asked for a time that wasn't available, the cast member didn't give me any alternatives and said there was nothing available that day. I asked her to check again and she found something but wouldn't book it for me because I have a split stay. Some cast members will go out of their way for you, some definitely won't. Keep trying. It took me 5 hours and several phone calls (from the UK) to finally get someone to book my ADRs. I still have 5 more ADRs to make so I know I'm going to have to go through this again tomorrow and the next day and the next.....
Anyway, there are some really wonderful cast members out there,but sometimes it takes a while to find one. And if HS really is fully booked, could you try playing around with dates or times? Maybe book a late lunch instead?
Good luck, whatever you decide.