Many Teppanyaki restaurants are mainly beef.
I have never been there, this restaurant seems to be selling vegetables.
There are also pork and chicken.
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