You don't think it could really torque the system board every which-way?

No, I don't believe it will. Though I wouldn't be sure in case you'd have to make an emergency stop.
Especially in the case of an AMD processor (which I believe you're using, right?) Those CPU's have VERY fragile cores. Putting a normal cooler on the wrong way is enough to crack it, so I can imagine that a cooler of more than 750g of weight can do some serious damage.
I would really consider removing the cooler before transporting the unit. After all, removing it and putting it back on is an easy enough process.

Could there be any problems from having the heatsink very close to the PSU shelf?

I don't see why it should. As long as the PSU and the cooler aren't touching each other and the PSU doesn't blow hot air onto the CPU you're golden. I wouldn't worry about it.