"Reverse etch" PCB base(not the copper)
You really, really, want to assemble the circuit on a thinner (or flexible) PCB.  If you have gerbers or similar for the PCB, just send them off to a PCB manufacturer and get some thin PCBs made.

If you already have (or are buying in) a small number of single-sided PCBs and need to reduce their weight then you could try grinding down the back. Use a metalographic polisher/grinder to remove the fibreglass - go slowly and use flood cooling. If you're careful, you might get down to 0.1mm or so.  You'll also be grinding away most of the legs of any through-hole components so you'll need to check that they haven't come loose afterwards. That won't work on a double sided PCB, obviously. 

Most PCBs are a mixture of epoxy resin and glass fibres. You could probably find a strong acid to dissolve the epoxy. The only thing which would dissolve the glass would be HF.  Both of those will destroy most components, and getting even a few drops of HF on your hand will kill you.

I very much doubt you'll find a chemical agent which will remove the fibreglass and leave the copper and components undamaged. And even if you did, the end result would probably fall apart before you got it onto the new substrate.