My dad bad-mouths the favored team while watching sports.
My dad talks bad about the favored team while watching sports.
My dad talks shit about the favored team while watching sports.
ご質問ありがとうございます。
「悪口」はそのままにbadとmouthに訳すのが自然ですが他の言い方はtalk bad aboutとかtalk shit aboutなどです。shitは砕けた話し方ですのでご注意ください。
「贔屓」はfavorと言いますので、「贔屓チーム」は受身形でfavored teamになります。特にスポーツの話ではthe favoritesだけでも通じられます。例：My dad talks bad about the favorites while watching sports.
ご参考いただければ幸いです。
He watches the game while bad-mouthing his favorite team.
・He watches the game while bad-mouthing his favorite team.
・He keeps complaining about his own team during the game.
・He watches the match grumbling about the team he supports.
・He roots for the team but keeps criticizing them the whole time.
1. **bad-mouth one’s favorite team**
“bad-mouth” は「悪口を言う」「けなす」という口語表現です。
例: *He always bad-mouths his favorite team when they make mistakes.*
（彼は贔屓チームがミスすると、いつも悪口を言う。）
アメリカ合衆国 
