Ask HN: Appointment Booking Software?

One of the famous HN users patio11 runs a similar company. Check it out

I need something that works in Australia.

Outlook?

SMS reminders are in the making and the rest is already available. I'm not sure, though, what you mean by "Calendar invite".

Disclaimer: I know some of the people who built this and haven't actually used their product.

Disclaimer: I know some people who work at this company but have never used their product.

BookingBug is aimed at this space. I don't know whether it has calendar integration.