Regarding how I obtained British citizenship

Hey guys I have a question about my own British nationality and how I obtained it.
So l'm suppose to have gotten my British nationality through British Citizen by virtue of section 1(1) (b) of the British Nationality Act 1981. - supposedly by my father .
My father gained ILR on a different nationality passport in 1988 and after a few years his passport expired. He then changed over to a German EU passport in the U.K. (issued in the U.K 1990.) and has used a German passport ever since. (My father lost his passport which had the ILR stamp)
After this he sponsored my mother via EU dependent visa from his German passport in 1994, my mother came and 1 month after I was born she gained her ILR through my dad's Eu dependent visa.
So when it came to applying for my very first passport when I was born, how did my father use his German passport and my birth certificate to get my first British citizenship passport?
Does this mean that the German passport may have some sort of ILR information or did they use his German passport to check his immigration history ? Or did I get my passport through him exercising treaty rights? Or did he automatically get permenant residency because of residing continuously for 5 years in U.K.? Which could have showed via passport?
If that is the case, surely me submitting my father's EU german passport which was used around the time of my birth should help my child's British passport application Then ?
(Hm passport office have mentioned it was from his ILR in 1988, but I'm trying to figure out is how they gathered this that's why I'm asking) - they are now asking me to show he exercised his treaty rights because I am unable to give the ILR from 1988.
I'm just trying to figure out what happened so I can understand how to deal with my child's passport application, if anyone has some sort of idea I would be grateful!

It is hard to know what your father did - have you tried asking him/your mum?
As he was an EEA citizen he would not have had ILR in 1988 but rather acquired permanent residence under treaty rights. The way to prove that would be to provide evidence that your father had exercised treaty rights for 5 years prior to your birth (eg by providing P60s for his employment). So to prove your citizenship (in order to prove that your son is also a British citizen) you need to provide your entitlement to British citizenship by proving your father had exercised treaty rights for five years at the time of your birth, your birth certificate, your parents' marriage certificate, then you and your wife's marriage certificate and your son's birth certificate.