A3 is the "problem" sounds correct.
Just in general, a battery box uses a resistor and a capacitor in its circuit. The values of these components, together with the input impedance of the recorder will determine where there is a bass rolloff and whether it is outside of the audible range (that is, roll-off occurring at below say 20Hz).
The line input of the LS10 has in input impedance of 78,000 ohms for the line in (2,000 ohms for the mic in). In comparison, the Sony M10 has a line input impedance of 22,000 ohms.
With these differences and depending on the resistor and capacitor values for the A3, you could end up with the bass roll-off of the A3 plugged into the LS10 at say 40Hz (meaning, signal -3db down at 40Hz and -9db down at 20Hz). Which isn't much roll-off, at least when considering concert recording where there is often an awful lot of low frequency content -- subwoofers cranked way up.
Plug the A3 into the M10 with an input impedance of 22,000 Hz and the bass roll-off occurs at closer to 80Hz -- meaning -3db down at 80Hz, -9db down at 40Hz, and -15db down at 20Hz.
These are made up numbers here, but in the realm of possibility, with the actual numbers depending on the specifications and components used in the A3.
But it is definitely possible given the differences in the line input impedance of the M10 compared to the input impedance of the of the LS10 that using the A3 battery box will result in a noticeable bass roll-off when used with the M10 that isn't heard when using the LS10.