Here's another suggestion, all BA flights on one date, tried three random dates in April and May:
1. LHR-AMS BA428 06.40-09.05
2. AMS-LCY BA8496 09.55-10.00
3. LCY-DUS BA 3273 11.00-13.15
4. DUS-LHR BA939 14.00-14.30
5. LHR-EDI BA1448 15.30-16.55
6. EDI-LHR BA1463 19.25-20.50
7. LHR-ABV BA083 22.35-04.50
The itinerary is not overly attractive, but rules said nothing of "need to fly" to prove. No margins for weather irrops but maybe the last two flights. No extra time zone bonus used. No success in trying to fit LGW in, but six airports ticked off despite that.