Hi !
Bootstrap just a framework to build the front-end, it has nothing to do with the engine running on back of your website. So I think what you need for the back-end are PHP to build login / registration system, and a mySQL database to store membership data. For the front-end, I'd propose lightweight, native HTML, CSS and javascript (and probably some images when necessary). No Wordpress please :)
What I did a few years back was to build a basic login / registration system which has a description below. It was a website selling an online chart for families:
(a) Any person may apply membership through registration process. Admin should approve his / her application first before this person become (accepted) member.
(b) When accepted, member may have choices whether to buy chart for one, two or three years usage. (S)he could pay this via Paypal or other payment methods.
(c) So whenever this member logins, (s)he will see that his / her chart is activated and is ready for use. After a year (or more, depending of type of chart bought), this chart shall expire and (s)he may want to extend it by paying again through Paypal. A month before expiration date, both admin and member shall get notification.
How does that sound - does it sound similar to the scheme you described here for this project? Kindly let me know,
Hendra D.