The three pins that make up the "XLR" are not always wired the same.
Consult the makers.
Consult the makers.
Perhaps you have ground loop. RCA cables' shields are normally grounded on one side only, while XLR cables shields are grounded on both sides. It makes for better, shorter return of the noise induced currents but creates a lot of ground loop problems in recording studios that exclusively use XLR. I would try to connect together and to the earth ground all the chassis in star fashion (no daisy chaining) but I'm no "Hum Whisperer"
I am (was?) battling with phono 60 hz hum in my system, too. Spent a lot of time swapping out components, star grounding, floating, relocating, cleaning connections, etc, with no luck. Yesterday may have been an epiphany (not 100% certain just yet). I think it may be a defective IC (RCA). This particular IC was previously defective (humming and shorting out; OMG the loud, crude noise it could cause if wiggled the wrong way), and I had it "repaired" at the factory. I reinserted it in my system concurrent with upgrading my phono stage, got the hum from the get-go, and assumed it had to be the phono stage I had just acquired. Finally, yesterday, I swapped out the IC with an identical model of the same manufacturer and the hum appears to be gone. Now, to be certain, I need to reinstall the aforementioned phono stage and see if the hum remains gone (I had used a full function pre to do my most recent sleuthing, which also had hum, so I figured the phono stage is not necessarily the culprit). My belated point is this: In my experience, ICs can and do go bad, and can cause hum or just plain short out. They are, IMHO, part of the investigative process and should be evaluated and tested when tracking down a hum.
Just got back from much needed vacation and just catching up on all the responses. The XLR cables are from Signal Cable and I spoke with the cable builder Frank and he assured me that the cables are wired correctly. Now of course I could switch to single ended cables (RCA) but I'd like to keep the XLR cables if possible. I'm going to try and lift pin 1 on the input side of one of the cables and see if that works. I'll keep everyone updated.
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?