Include HTML template file in wp_mail

Per my comment to your question, I believe the problem is that includeing files, whether directly or using get_template_part isn’t likely to give you a string to pass to $body and that is going to cause errors in the code, or at the very least unespected behavior.

I would avoid reading files into memory and just create a function that returns your $body content.

function get_email_body_wpse_96357() {
    $body = '<p>Hi</p>';
    return $body;
}

Then use $body = get_email_body_wpse_96357(); as needed. An advantage of this method is that you can easily pass parameters to the function if you ever decide to do so. You could also use variables in an included file but it can be messy.

If you don’t want to load that function all the time, then put it in a file by itself and include that file only when you need the function.