I considered using the 144 hour transit visa for a trip to HK with a stop in Shanghai, but then needed a visa anyway for a trip a couple of months before (and they gave me a 2 year multi entry which was nice and meant I didn't need to try it.). I was curious though about this visa option and thought it might come in handy another time. So on departure for the first trip I asked the BA checkin staff whether they were familiar with this visa and was told that they had heard of it but the checkin person might have to make a call to confirm entitlement. On arrival at PVG l checked out the Immigration lanes/booths and there was one dedicated to transit visas.
Beijing Capital Airport suggest it is straight forward
Beijing Capital Airport 144 hour transit visa
But as you plan to use Air China why not speak to them? They should know.