I wouldn't use it because there's no way that adding two new plates (and 2 new gaps) could possibly increase thermal dissipation. If your heatsink isn't applying enough pressure just increase the bend on the two plates that apply the pressure, the bend will flatten out a little every time they're removed so you really should do it each time.
copper is great at conducting heat. By adding in the plate you create a new gap between the copper plate and the heatsink it attaches to, the gap may be filled in obviously but it's these gaps (or joins depending on how you look at it) where the thermal efficiency takes a nose dive (and why we use thermal paste).
The only to way switch to copper to increase the efficiency would be to replace the entire heatsink with a copper one not just a thin layer of copper on the bottom.