One CiviCRM for multiple sites
I would always recommend to have CiviCRM on a separate server from the public facing website (privacy, security and updates) even thought it is more cumbersome and more expensive.

So I assume you will have 3 WP websites, and one central CiviCRM (in a dummy Drupal or WP). I would use CiviProxy to ensure only whitelisted API calls are accepted (check https://github.com/systopia/CiviProxy for a little more info) and then there are a few options I think:


Ensure your WP sites can process the data and prepare an API call to CiviCRM using the core API's
Use the work done on Gravity Forms with CiviCRM although I am not sure that will work with REST
Just throw all the data on a Wordpress form to CiviCRM and create a customized extension that provides specific API calls to process the data. We have done this with the CiviCRM Contact Form 7 plugin and specific API's.