Our new puppy took the place of our old dog after it died.
Our previous dog was replaced by a new puppy after it died.
ご質問ありがとうございます。
「lose out to」という表現は「負ける」という意味でなので今回の場面で使えないと思います。
「先代の犬は死に主役の座は新しい子に奪われた」は英訳すれば「Our new puppy took the place of our old dog after it died. 」になります。
別の言い方で「Our previous dog was replaced by a new puppy after it died. 」にも言えます。
一つ目の英文と同じ内容を伝えられます。
アメリカ合衆国 
