He was a selfish father, but he died without causing a problem.
He was a self-centered father, but he died without making a fuss.
ご質問ありがとうございます。
「わがままなXX」はselfish XXとself-centered XXと言います。
「迷惑かける」はcause a problemとかmake a fussと言います。make a fussはcause a problemよりカジュアルな言い方です。
上記の回答では書いていませんが、「最後は」はin the endと言えます。
例文:He was a selfish father, but in the end, he died without making a fuss.
ご参考いただければ幸いです。