I've loaded my civic with many other Honda motors before, and besides a slightly dented trunk floor, it handled it just fine. A new clutch might help though.
On my civic, it is lowered, and the higher-than-stock spring rates help it handle decently and not bottom out even with a lot of extra weight in it. I've gone with Ground Control coilovers. You can raise them up to factory height as well. If you went with some stiffer springs threw that route, it would help. If you want to go factory, The integra has the same rear suspension components as the VX but has stiffer springs. It will probably be a bit taller though once installed. You can compensate by cutting off a little probably, though I don't have experience with cutting factory Honda springs.
Another route you could take if you want to use stiffer OEM springs, to compensate for the added height they will give the car, is to install struts with adjustable perches. The Koni Sport struts I have allow you to raise or lower the height of the perch for the OEM style springs.