I just tested a combination of hori 6, ave 6, and the RX 6 button with two different taps. One of my taps is geeky - because if you just breathe on it with a 2nd pad plugged in - it starts to wig out (random button presses). It doesn't matter which port the 2nd pad is plugged into. Probably needs a re-cap on the tap. It has the behavior sort of what your described. And when that does happen, I have to switch the 2nd pad back to 2button and then back again to 6 button to fix it (or sometimes reset the game). As in the game logic detection gets in a weird state (because it can dynamically change back and forth between the two configurations).
I tried with my 2nd tap and it was rock solid - no issues (I smacked the hell out it if too). I don't have any mods on the system, but I am running a TeD Pro is which does draw some current over stock. A tap with extra pads it going to as well.
Try re-capping your TAPs. Not sure if the 6 button controllers have any caps.
FYI -
The 6 button uses a counter as a flip flop circuit, so the odd reads of the same controller gives a directional input reading of ALL directions (an impossible state). It's how it knows what is a stock "read", and which read is for the extra buttons (you look at the directional state when looking at the button states). I mean, this does require an extra IC in the controller two work properly when in 6 button mode. I only have 1 AVE-6 pad, so I can try two of those specific ones at the same time.