As an experiment, temporarily try connecting BOTH the RCA and XLR cables, with the input select switch on the BAT amp set for XLR.
Doing that might not be sonically optimal, but if it eliminates the hum it would help to confirm a possible explanation that occurs to me. I'm thinking that the cause of the problem may be an incompatibility between the design of the XLR output circuit in the Integra and the XLR input circuit in the BAT.
Specifically, perhaps the Integra's XLR output is actually single-ended, with a signal provided on pin 2, but pin 3 as well as pin 1 connected to its circuit ground. The BAT's input is presumably designed to respond to the voltage difference between signals provided to it on pins 2 and 3. And more likely than not the BAT connects pin 1 to chassis, rather than to circuit ground, that being proper practice even though many components having balanced inputs improperly connect pin 1 to circuit ground.
In that situation there would be no low impedance path between the circuit grounds of the two components, yet the BAT amp would be trying to sense the difference between the signal on pin 2 and the Integra's circuit ground that may be connected to it on pin 3. I believe the result of that may very well be hum. Connecting the RCA cables between the two components while utilizing the XLR connections would provide a low impedance connection between the circuit grounds of the two components.
All of that is just a guess, of course, but it's an experiment that's easy enough to try.
A separate question: By any chance is anything connected to the outputs of the BAT besides a pair of passive speakers? Such as a powered subwoofer, or a hybrid electrostatic speaker that includes a dynamic bass driver having its own amplifier? If so, considering that the BAT probably has balanced outputs that might be a relevant factor.
Finally, I'm not sure that a model VT-6200 exists. Did you mean VK-6200?
Regards,
-- Al
