WordPress or custom coding better for this project?

I’m a LAMP developer with a ton of experience building custom-coded websites in a variety of PHP frameworks who has been tasked to develop a website for a customer who strongly prefers that I use WordPress. My WordPress knowledge is extremely limited. I have built a couple of simple sites using Envato themes that didn’t really require any custom plugin development, or meddling with source code.

This site would be different. It would be a custom fitness tracking website where users can login, view videos of exercises, and log their workouts. (Yes, I know there are a million of these out there, but this company wants their own).

Here’s an overview of my major requirements:

• Customer accounts with secure login
• Paid Subscriptions with multiple levels allowing access to different features
• Videos of exercises
• Forms to log progress (time, reps, etc.)
• Charts and graphs to view progress
• Leaderboard to show comparisons of members
• Ability for future iOS/Android apps to integrate with the website/database

From my limited knowledge of WordPress, I would have to build some custom plugins for this. I don’t see anything out there that would accomplish all of these requirements. I’m also assuming that I would need a separate MySQL database to store the majority of this data.

Are these requirements able to be accomplished with WordPress in a relatively straightforward manner? How would a WordPress expert go about tackling this project?