144-h layover: London- Beijing- HK- back to London (on a French Passport)

Hi, my Bf and I are planning a trip to Asia for Christmas to visit his parents in HK.
We would like to fly from London to Beijing, stay there five days and then go to HK for a few days before going back to London. I am travelling on a French passport (although I live in London), can I take advantage of the 144-hour layover in Beijing?

if yes, can I claim the luggage in Beijing?

Kind regards,
Julie