We don't have an approval process for booking on events. We do have one for new members, but it's a one-off approval, not something that could be used repeatedly for different events.
However, here's a suggestion for what you could do:
Create 2 events. Make the first event a free event (and add any forms to collect any information you need from applicants etc). Once you have selected those have a place, you can send them an email from the first event booking screen, inviting them to the second event.
You can't prevent people from booking on the second event who weren't invited. However, if you are linking to individual events from your main site, you could link to the first, but not to the second, making it harder for people to find it. Also put something in the event description in bold text that it's an invitation only event.
