I think the mystery of the Moller gap has finally been solved.
Looking at the reconstructed Moller events generated with the fixed cross section, not much was changed,
so I explicitly forced the condition such that if one Moller had track E > Ebeam/2, then the other had to have E < Ebeam/2, and vice versa.
Applying this to both MC and Data, along with modest ESum and phi cuts, gives the attached momentum plots.
The other distributions match much better too, particularly track position at the ECal, which now shows the gap in data as well. The higher-energy bias for hits in the bottom half of the ECal can be seen from the asymmetry in where the electrons hit. Loosening the MC cuts a little (only cutting ESum) starts to close the gap and make it look even more like the data plots.
So in summary, the gap is likely due to electrons always being cleanly paired on opposite halves of Ebeam/2 in MC, but not necessarily in data. The gap is also apparent in background MC without forcing this condition, suggesting that the MC is too "clean".