Lifespeed, I am not having the same cornering issues with my 04CVT HCH.

I have the factory stock Dunlops @ 50PSI, very hard tires.
In my daily route there are a few turns which I simply REFUSE to loose built up momentum and therefore take the turns rather hard.
When my family is with me I am more carful but still they've learned to "hang on".

2 Turns that come to mind in paticular....
It is a downhill approach to a right turn lane....but at 2:45 in the morning I take the turn WIDE, hard and fast, and I can hear the tires complaining as louder road noise but not screeching...
Another hard turn is taken near to my subdivision at 3:00AM going ~30MPH without my engine running for a mile. I take a sudden hard left and once again the tires get loud but not screeching...all instances I have no loss of control.

I'm not sure how heavy the assembly is behind the rear seat but I'm not experiencing what you describe even with my family along.
Civic's have double wishbone rear suspensions, which is about the best setup.
Civic's also have a F/R stabilizer bar.
Another thing you might want to have checked is the shocks/struts.

Efficient drivers do it better.
1003 miles a tank personal record. 74MPG calculated. HCH1 CVT