I don't understand how this could be a wattage issue, since this is an
auditable difference at low volume levels.

"Deep bass" tends to require more power to reproduce. This follows the 1/f
"rule". If there were differences at lower volume levels, then one of four
things must be true: 1) tone controls and boosts were active in one case but
not the other; 2) the difference in gain settings were causing a difference;
3) one of the amps was faulty; 4) the power of suggestion is powerful.

A few years ago I was running a
1200 watt 4 channel Pyramid bridged in to a pair of 4ohm 12's and it
powered
them nice and loud, but there just wasn't any low bass at low volume
levels.
Then I replaced it with a Kicker IX704 amp which had loads of grindage at
low
volume levels. But oddly enough it did not hit as hard at high volume
levels.
It's not just the Pyramid, I have a pair of Profile amps here that act the
same way.

Have you measured their frequency response? If something's wrong with that,
including at low volume levels as you suggested, then it's time to send them
in for repair.