Now when i see your measured values .
Lamp unit has somekind of voltage regulator ,maybe not only resistor, when you measured 3.3V and 6V lamp was connected
and shows brigth light. ( I guess your meter show some DC value, but there can be PWM regulator circuit ie. chopped DC-current
not pure ac and not pure dc).
Both batteries give about same power (P=U*I) 10VA
Because we know nothing about the circuit ,one reason can be that 4.7Ah battery has higher voltage, causing higher power dissipation in regulator.
Stabilating circuit can be also current regulator.
If additional heating is caused by higher voltage, it can be tested by connecting 2pcs 2A diodes in series to lamp unit ( this makes inputted voltage 1.5V lower )
or take out one cell.
KAK