Can I add an upgrade to private coaching as a program on site and they pay
right there?

Yes, you can add coaching to your programs and to your site and it will process through the site as any other program would.  For example, you can set Group 1 to be the program just as is and then Group 2 is the program with coaching.  You can also create an entirely new product and sales page for your private coaching and it will still process through your site as the other programs do.