Expand heatsink contact area with larger copper plate?
This sort of thing is reasonably standard in the RF world where the usual problem is small geometry power devices mounted onto very large typically aluminium heatsinks, power levels can exceed 1,000W. We typically machine a copper block then solder or (vastly inferior) screw the power device to it in order to increase the contact area between the heatsink and the spreader by many times over the contact area of the raw device. 

The phrase to search for is "Heat Spreader", and I would note that surface finish is reasonably critical, face milling of both components is commonplace. 

Your case is slightly the reverse as your loads are bigger then the heatsink area, I would go for a bit of 10mm MIC-6 or similar ally tooling plate (Copper is not really that useful to you as you do not really have the power density to make it worthwhile and it is a pig to machine), tooling plate is nice as it has already been face milled so will be reasonably flat, something that cannot be said of extruded bar stock.

Another option would be a single sided metal core PCB made with a very thick substrate as a heat spreader.

TDP is not really a useful number, being as it combines a mess of thermal resistances and thermal limits (which may or may not apply to an LED), far better to use the thermal resistance (measured in kelvin per watt) which makes doing sums to see how hot the die will actually get fairly straightforward.