Build me an online booking platform, or use an existing platform and create what I need.

I need the booking platform to handle online bookings for multiple attendees per time slot (eg 4 people per 15 minute block), I need to be able to adjust the appointment availability from 3 per 15 minute time slot to 5 per 15 minute time slot, private individual log ins / access or a separate booking link for each company (I need many different companies to be able to book their clients in for my service but I don't want them to see the other users on the platform), a waiting list if a time slot is fully booked, taking customer's email address & phone number at time of booking so I can follow them up or so they can be notified when a time slot is available if they are on a wait list, sms reminders to be sent out prior to the appointment, the ability to receive payments upon booking so it's either connected to Stripe or Square or Paypal or some other type of payment gateway. I don't mind if this is hosted on my own Wordpress site, or in the cloud (air?), or if it's a stand alone piece of software. I am open to suggestions.