Someone on H-T was doing a lot with the CRV diff, and I recall shims & added pressure is all it needs. IIRC he shimmed it, sealed the relief valve, and used a seperate diff pump to create pressure & cool the fluid.
The front has to spin a bit before the rear diff engages. The heat is just from the slipping, which is from intentionally not being shimmed tight. The way that diff works is when the front wheels slip, and therefor turn faster than the rear wheels, the pump starts moving and adds pressure to the clutch discs. When the front wheels spin the same speed, the pump doesn't spin/create pressure. It almost can't function when the rear gets grip, that's why the RT4WD diff is better for getting the AWD effect. That system used the viscous coupler to have it always engaged to some degree.