3 Days in Shanghai

No, unfortunately your itinerary is not eligible because you have two mainland Chinese cities in a row. I understand that you see Beijing as just a transit city for your itinerary, but the rules of 144 hour visa free transit are clear and they say that the immediate points before and after your Chinese destination have to be in two different countries different than mainland China.

If you cannot modify your itinerary, then you will have to obtain a Chinese visa in advance.