^I hope the Velo experts have suggestions for you, but here's my take.
My thought is that as you ran the Velo using the speaker connections with no problem (IOW you cannot reproduce the problem using that connection) then IMO you have narrowed it down to somewhere in the preout path. As restarting the Mar solves the problem, that kinda points to the AVP-have you posted in that thread?
In any case I'd want to really narrow it down further before sending the AVP in for service. IOW it could be the AVP sub channel electronics, the AVP RCA output jack, or possibly even the RCA sub cable connection so I'd want to determine at what point exactly in the chain the signal is missing or lost.
What I'd do the next time it occurs is try some more diagnostics, establishing whether the AVP is outputting sub ch signal. I'd unplug the sub cable from the AVP and touch the end. The sub should hum/buzz each time you touch the end, telling you the sub is fine.
Next I'd test for signal at the sub preout using a multimeter. This might be easier if you connect a short known good RCA cable there. If there's no signal, jiggle the short cable a bit. if no luck, do the AVP on/off cycle and check again. If that fixes it, it's internal to the AVP.
This doesn't sound like a bass mgmt settings issue but I'd start with have speakers set to small with an 80 Hz crossover. Also, what value does the AVR autosetup set the sub ch output to? I'm also curious whether this problem happens only with 2 ch (2.0) sources or with sources with distinct LFE content (5.1).