You can book the BBB appointment at 180 days + the length of your stay (same as dining reservations). So if you are staying for 5 days, it’s 180+5. For 10 days, 180+10.
I don’t think 3 PM is too early, unless you want your appointment as close to the party as possible (the hairstyle might hold slightly better, although you can also bring along some hair spray and use that after your appointment). You could ask for the latest appointment they have. If you wanted the appointment as close to the party as possible, I might do that. For comparison, my family usually has a pre-party dinner at Be Our Guest. I try to look for a 4:30 reservation for that.
Do you have to call for tickets to the party? I know in the US we don’t. I can buy those tickets online. BBB, unfortunately, has to be booked on the phone.