I had similar issue, I have three last names (Father, Mother and Husband's), therefore my full name is very long, I provided all the versions some organizations have used here in Canada and also checked my Credit report as you can see what are the names they have for you and reported all of them in my application as Alias/Nickname. In your case as your father's last name could be use as Middle name, I would suggest that you write a letter of explanation, indicating exactly what you have stated here about your Father's last name and include that with your application.Hello, I have a question, about how to fill the other names used section.
Name at birth will be my current name, now where I am from there is a culture to use fathers name as middle name, hence some of my documents have my fathers name as middle name, so when I provide that information, it asks me why was this name used and the options are 1) Name at Birth, 2) Maiden Name, 3) Married name, 4) Previously married name, 5) Alias/Nickname.
Not sure what to select.