If the piece of metal between the magnetic pickup and the string is non-magnetic then the
'communication' between the string and pickup will not be affected.
But, and this is only my opinion, the outer ring of the pickup cover and to some extent the
rest of the cover will act like a 'shorted turn' since, whilst it may not be magnetic, it
will still conduct electricity.
This 'shorted turn' on the pickup winding will reduce the 'Q' of the pickup and any 'resonances'
due to its inductance and self capacitance.
So it will sound different.
Channelman