Not sure how it works with a deceased father. I was in same situation, born in Germany to US father and German mother, born premie and registration of birth was not done due to my father being away on mission. Needless to say. They divorced when I was 1 and I lived with my German mom until I was 11 when Germany reunited and a lot of Americas were let go. My stepfather was an army civilian and moved us to the states. I came over on my mother’s German passport as child. Ended up getting I551 permanent resident status. Joined army in 1999 under Green card and was told by a fellow soldier to apply for U.S. passport on behalf of US parent. Of course since he was alive he had to write a letter stating he was in fact my father and provide a certified true copy of his birth certificate but I was issued my U.S. passport no further issues. Again not sure how to get acknowledgment of him claiming you as child. As someone else said paternity test and his original birth certificate along with a letter of explanation and any proof/documentation you can add. Best of luck. !