Pain in the R's
The McDonalds next to the SEN Holiday Inn operates H24 so that would seem to be a reasonable place to wait, for whatever reason, during the terminal closure period of 00:00-04:00. If the last train to London has gone Andrews Airport Cars offer transport to Liverpool Street and other points for around £25 per head in their seven-seater vehicles (obviously more is charged if only a couple of people need the service).
I assume your comment about closing the terminal for part of the day in winter is tongue in cheek as that would obviously not be practical.
I will bow to your superior knowledge on the matter if no delayed arrivals have previously landed after 02:00 but I seem to recall it happening at least once. ATC is H24 and RFF cover is Cat 6 H24 also so it is only a matter of airfield operations providing sufficient staffing to cater for late arrivals. Whether or not easyJet prefer a diversion to STN rather than paying any additional handling costs at SEN is a matter for their commercial judgement but I would have thought the latter would be more sensible, if available.