Your code is executed in session 0 when hosted on IIS, so you can see the process in Windows Task Manager but cannot see its UI elements (session 0 has an invisible desktop).
There isn't any workaround for you with
Process.Start, as the modern web browser security policies prevent a web app from creating processes on the client machine (high security risk).
If you do need more guidance on your specific scenario(s), you might hire an experienced consultant or contact Microsoft support. After thorough there might be alternative design approaches that fulfill your requirements.