Windows authentication popup of my application appears similar to this app https://www.engprod-charter.net/ I had used below code to handle window authentication popup in se
This can be easily done using Robot Class in java, which can be run in Robotframework test when required.
i.e. A .jar which will function as handler for the windows authentication pop up.
In RobotFramework you would just need to run a .jar which will handle your pop up.
Just in case you are not used to Java, I can help you with a sample code.