144 hour to Chengdu with a 3 hour transfer in Beijing

The basic rules of 144 hour visa free transit say:

- The immediate points before and after your mainland Chinese destination have to be in two different countries
- It is not allowed to have two mainland Chinese stops of any kind in a row.

As you can see, you cannot proceed with your itinerary without a visa obtained in advance because you will have two mainland Chinese stops in a row. The problem here is in the fact that in China, like in many other countries, the formalities (immigration, customs...) are performed at the first stop. In other words, you have already entered China on your first stop and it is not possible to stay airside because your next flight is purely a domestic one.

For example, your second segment is fine for 144 hour visa free transit. I suggest you to try to adapt your itinerary to be in accordance with 144 hour visa free transit rules or to obtain a Chinese visa in advance. I'm sorry.