Have you confirmed that earth ground and case ground are good?
When I get noise on anything I check this first.
Good point. These PSUs are normally very well grounded to the motherboard ground. OTOH, you want to be careful about running multiple GNDs to different places. It could create noisy ground loops. Maybe that is the place to start.
An Amiga will draw only a fraction of the 450 watts available. I guess it's possible that it is not fully loading the PSU to the point of being stable.
Most scopes have very high impedance inputs, so as not to load the circuit under test. I don't see why you could not directly monitor the 12VDC rail with it.