Sam, you need an earth bond on that copper screen in order that it may be an effective RFI / electrostatic shield. Sorry if I missed it from your picture but others may miss it also.
Of course it will prevent voltages from being capacitively coupled, but will do nothing to prevent magnetic flux leakage.
1/4" thickness of ferrous material will give about 20dB of attenuation.
Dont confuse the copper band on the out side of some transformers with your method of shielding. The copper band is continuous and caries and eddy current which in turn creates a flux in opposition to the main leakage flux thus offering some attenuation. A copper electrostatic shield some times placed between windings is not continuous, it always has a gap else it would cancel some of the main magnetising flux and reduce the transformers efficiency.
The core will often leak more at some points than others, usually at cable entry points so careful positioning and rotation can help. But I think this level of detail is normaly only required on instrumentation work.
However your R-core has about 1/10th flux leakage of a conventional E-I core so the R-core has effectively moved the its self three times further away than an EI-core. I think... (not doing the squared law very good this morning)
Edit, its the cube law not square law.