Your subs are not getting enough level. The level they are getting is so low that unless you are playing loud it is below threshold and that is why they shut off. This is also why they don't play loud no matter what you do with the volume on the subs. When the input is too low no amount of turning it up will ever be enough. They only have so much gain. You can forget about cables and splitters and how they are wired, your problem is line level.
You can check this with a meter across the RCA while playing music or a test tone. Line level should be around 1v to 3v. In your case it is probably a tenth or less of that.
A big clue by the way, this all started with the new integrated. If you can't get enough signal from its pre-out the next move would be to run the subs off speaker level using speaker terminals on the amp. If the subs don't have speaker level input then you get a $20 Line Out Converter.
That amp is plastered with doo dads and features. If you are really lucky you just need to change a setting to allow pre-out. Wouldn't that be something if that was all it was?
