The translation of ‘I love her’ is not necessarily problematic, but is not a direct translation. ‘The girl who I love’ or ‘Her, whom I love’, or ‘My beloved’ would be closer. While 私 is simply omitted in the original, it could be included to state 私の大好きな彼女.
According to Wikipedia, in this anime, the female lead is referred to only as 彼女 (her). The cat is in love with his owner and seems to consider ‘her’ to be his ‘female partner’. Apparently, this anime anthropomorphizes the cat specifically through the eyes of the creator and was made for the girl he loved in real life.
If ‘her’ name was known, and it was のりこ, the cat would say 「（私の）大好きなのりこ、私は彼女の猫だ。」 instead.
To make it simpler, take the expression 好きな人, which means ‘someone who is liked’ and apply this to the expression in the anime.
For those of you viewing the source video, the line in question is at 2:18.
Side note: In your post, you offer 彼女を大好きな as an alternative, which is not grammatical. 彼女が大好きな＿＿＿ or 彼女の好きな＿＿＿ would be fine. Adding な after 好き makes whatever comes before 好き an indirect object, so を (indicating direct object) wouldn't jive.