I recommend Fuzhongjia saute spicy chicken with a large private room. The food is good and not too expensive.
Never go to Xiangpaifang hot pot, it's super expensive. Little sheep is better, but it's also expensive.
Other dishes in the south gate, such as Zhongyuan Food City, are just so-so. There are also some small hotels, and there are basically no big private rooms.
The Jijia restaurant in Ximen is very distinctive, delicious and large, but there are few dishes.
In fact, there is another place, that is, food stalls, which are very affordable and delicious, but it's cold now. Let's go in summer.
I hope it works for you!