Dennis, if you know of a machine shop, close by, then you can get the perspex polished on a buffing wheel. They should have some 'buffing soap' which is used as a lubricant/polish.
Can also use 'Brasso' for the final polish.
'We' did this years ago (Electrical Apprenticeship), when making 'foreign orders'....Errr, table lamps/lamp stands etc etc out of 1/4" perspex, (painted in diff colours, of course), interleaved with 1/16th" brass sheet on a 3/8" brass tube.(for the cable).
Also made knife handles n such.
Came up very nicely.
HTH.
L.