The Moodle default form cannot be overridden. The external fields also cannot be added to a Moodle Form.
While looking for a solution to overcome this, we at Edwiser figured out that we can actually replace a Moodle form with another form. We tried replacing the default Moodle Login Form with the ‘Login Form’ we created and it worked. Even tried customizing it and this worked too. So replacing a Moodle form can be one solution for it.
Hope this helps what you are trying to achieve.