override the field you want to add option:
class YourClass(models.Model):
    _inherit = 'module.model'
field = fields.Selection(
    selection_add=[
        ("new_option", "New Option"),
    ]
)

How could I apply the activation of my new status when making a credit note?

Hi,

https://www.odoo.com/forum/help-1/add-value-in-selection-field-without-overwrite-whole-field-160683
Refer the above forum post to add a new option for the selection field in Odoo

And for updating the invoice status, you can use the _get_invoice_status(self) compute function which can be found from sale_order.py file in Sale module.

Regards