cheena First of all, It'll be always much cheaper and easier to run an already developed platform (as Flarum) than building one from scratch, as the developer said it would do.
If he wants to integrate the site, yes, he should use the API, using the API doesn't change in any way the functionality of the forum, it just lets something communicate with Flarum.
Installing Flarum, you can have either forum.mysite.com as mysite.com/dashboard/forum