Is London - Beijing - Hong Kong - Shanghai - London valid for Transit without visa?

I would very much like to fly from London to Beijing at the end of May, then after 3 nights (<72 hours) fly to Hong Kong for 4 days then to Shanghai for 4 days and then back to London.
I have read on the Shanghai Genaral Station of Immigration webpage on the new 144 hour TWOV procedure, and it seems clear that we will be able to use the TWOV rules in Beijing and Shanghai because we are flying to an external country (Hong Kong) between the 2 Chinese cities. Your website explains the visas well and how to use the new lanes at the airports, so I was about to purchase the flights, however UK travel agent TrailFinders is telling me I need a full Chinese Visa, which is very expensive (£175) including the checking and submitting fee for the CVASC. I'm worried that I've got it wrong now... Help!