Before ticketing you can do whatever you want.
After ticketing it's hard to get an agent to change a stopover to a connection and even if you do it may get rejected in the ticketing process which will be a headache to fix.
All the more complicated by the fact that the agents themselves never seem to know if a connection is 6 or 24 hours.
That being said I managed to do it several times, but it also involved jeopardizing the entire trip, but it can be done.
Updated itinerary (HT: Dan for suggesting changes)
July 31: American LAX-SFO 1st (M80)
July 31: Cathay SFO-HKG 1st (744)
August 2: Cathay HKG-DME Bus (343)
August 11: British DME-LHR Bus (767)
August 11: British LHR-CDG Bus (320)
August 31: British ZRH-LHR Bus (319)
August 31: British LHR-LAX 1st (744)
Total mileage (according to GCM): 19,432 miles flown
Note: I tried pushing to allow a 40 min layover in LHR to get from my DME to CDG flight on 8/10, but the agent told me she couldn't override the minimum 60 minute rule in the system. No worries though, because the flight on 8/11 is first thing in the morning anyway.
Thoughts, comments, suggestions, etc. welcome as always...