I could see drilling out flare fittings, but this cooler uses barb connections. There isn't anything to drill.

I wonder if the pressure from the pump is sufficient to increase the velocity of the fluid through the 5/16" barb connection to maintain the volume on return to the transmission?

The only point the system would be necked down from 3/8" to 5/16" would be the 2 one inch long barb connections.