Can I go Visa free?

Unfortunately, there is a problem and your itinerary is not eligible for any kind of visa free transit scheme.

The very basic rule of 144 hour visa free transit scheme clearly states that the immediate points before and after your mainland Chinese transit city, Beijing in your case, have to have its both immediate points, regardless of their nature, in two different countries or eligible territories (eligible territories are Hong Kong, Taiwan and Macau).

As you can see clearly see, both of your immediate points are in Hong Kong. Trust me, you will not be allowed to board the plane bound for Beijing if you retain an itinerary of this kind.

I suggest you to change your itinerary and to depart to Beijing by a direct flight from Macau. After your planned stay in Beijing based on 144 hour visa free transit, you can board the flight to London with a stop in Beijing.

Please, do not rely on the, obviously, weak knowledge of your agent. It is enough to check Timatic and everything is immediately clear.