I would not consider Berlin for a day trip. While TXL airport is reasonably close to the city (although keep in mind that Berlin very stretched out), access is only by coach/cab and traffic in the afternoon can be bad and security has become a big problem at TXL in the past year.
How about considering the not so obvious destinations, i.e. Rotterdam or Dusseldorf. To both destinations, you will not waste too much time on flying.
Dusseldorf airport is a quick 12 minute train ride into town. Most attractions are within easy walking distance, it has good shopping, art and excellent food and a nice scenery on the Rhine embankment.
https://www.planetware.com/tourist-a...f-d-nw-dus.htm
Rotterdam is a small, easy to navigate airport. The city has interesting architecture. For food, the new markethal is nice.
https://www.planetware.com/tourist-a...nl-zh-rott.htm
One other aspect you should consider is that most BA aircraft do not overnight at LCY, so the first outbound flights to most destinations are somewhat later in the morning. If you add to that the time difference with the continent, at most destinations arrival will not be before noon. Hence my suggestion to look for a destination not too far away as the aircraft will arrive from there at LCY first thing in the morning and one of the earliest departures from LCY, allowing you more time at your destination because of an earlier arrival of the first flight of the day (you will be at DUS as early as 0940 or RTM 0925, but at TXL not before 1105) and usually a later departure of the last flight of the day. Or go for a destination that is served by a LCY-based aircraft (e.g. EDI, AMS, LIN)