Probably the main contributor to the problems you are hearing is that the output impedance of the BAT and the input impedance of the DNA-500 are not a good match.
Stereophile's
measurements of the VK-51SE indicate a sharp rise in output impedance from 410 ohms across most of the audio band to 4700 ohms at 20Hz. That will result in a significant deep bass rolloff when driving the DNA-500's relatively low 10,000 ohm input impedance. Higher frequencies may also be affected, to the extent that the two impedances vary differently as a function of frequency.
I note, btw, that the input impedance of DNA-225 is far higher than that of the DNA-500, 100K instead of 10K. 100K is an excellent match for the BAT's output impedance.
Another factor might be that if overall gains and sensitivities in the system are such that you are using the BAT's volume control in the upper part of its range, that apparently causes its frequency response to deteriorate considerably. That occurs even when amplifier input impedance is high. See Figure 2 of the Stereophile measurements.
The only solution I can think of that does not involve changing or modifying components would be to add something like a
Burson AB160XLR balanced buffer between the two components. That would resolve the impedance matching issue. That particular buffer also apparently provides up to 6db of gain, so you would be using the BAT's volume control at somewhat lower settings, thereby helping the second problem I mentioned, if it is a problem.
Regards,
-- Al