I don't think it is as big a deal as you might think. Most castors are rated at 150 lbs each. you have to multiply that by 4. And that is the standard stuff. You can get heavier duty stuff at any Home Depot to replace stock units.
Find a stool you like, look at the castors. If they are cheapies, take the stool into the hardware store and replace them. Assuming the basic construction of the stool is sound then you are cruisin'!
I am a yard sale fanatic. I find old shop benches, stools, etc and replace the wheels, feet, etc, so I have done this a lot. Give it a shot.
:up
370lbs? I pity the guy that dings your door!