Although it is about 30 minutes away from Shinjuku, there are many stylish restaurants in Daikanyama.
I have been to a bar called "Tableaux Lounge". It is a jazz bar with a wonderful atmosphere. The music is played in live, and they will sing a birthday song for you if you ask them in advance.
Sorry I have not used any restaurants for birthday around Shinjuku.
I searched on Google.
Teppanyaki Orca - has dessert plate for birthday.
Kaiseki Kitaohji - good atmosphere
Many Teppanyaki restaurants are mainly beef.
I have never been there, this restaurant seems to be selling vegetables.
There are also pork and chicken.