I would be fairly confident the crossover is unbalanced inside...... And if it was mine, in your position I would be tempted to think about bypassing the input banacing stage, but it would probably make little difference to just running an unbalanced signal from a pre, if the lengths for that wire were sensible.
Anyway thought experiment shunt volume balanced pre, not too complicated. I am not as precious as some, I did work in BSS for a couple of years, one of the early pioneers of active crossovers in the UK, so I know what goes on inside these type of boxes.
A bit simplified, but something like this would work. Would use ECC88s personally or similar, and kep anode resistors fairly low, sub 10k, and it would need quiet stabilised power supplies.
would also be tempted to make output SRPP
or a PP output transformer for true balanced. Wherever, it could be made simpler or more complicated, but does mean you can use a normal stereo log pot, although it would not be as log... sometime we must suffer for our art.