To be honest in this sort of situation where all I wanted to do was boot the pc and update BIOS, assuming it fits I'd just leave the heatsink stuck on the 3200G and run it in the new mobo like that (if it won't attach properly to the new mobo, put the case on its side so gravity is weighing it down to avoid it falling off). You know you've got good contact between cpu and heatsink, so should be fine for the 10mins you need it for, you aren't going to be running anything cpu intensive.
Just seems like a bit of a faff to remove the heatsink only to have to put it back on again when you return it to the other system.