So make the pillar slightly taller than the depth of the hole, surface tension will suck the LED down minimizing the solder layer thickness where it matters. Everywhere else the solder only has to be electrically conductive which doesn't care about the joint thickness.
Is there an epoxy that could be used on the center pad & pillar that's better than solder? There doesn't have to be any solder at all on the center pad, but it has to have a good tight bond, whether that's from sheer interference fit/impingement, from epoxy, or from solder I don't really care.
The substrate has a thermal resistance of 2.5°C/W.
For a 1mm high post of copper (thermal conductivity of 400) with a diameter of 2mm I calculate 0.8°C/W.
Same piece made of solder would be 5.3°C/W. The substrate has a thermal conductivity of about 16W/mK. Nothing spectacular, but the area makes it work fine.
Since you only add 1 solder layer (one is needed anyways), you have to keep the new solder layer below 0.4mm.
Just some quick math made up from what I found on the internet. I dont know if the change from one surface to another adds much thermal resistance..
Indium might work. I think 80W/mK but you can solder it to about everything.
(off-topic) I did work with EM's half a lifetime ago, but we used glass knives for cutting slices. There was a fancy machine that could break pieces of glass so precise that the edge was dead straight and had exactly the right angle for slicing (well, most of the times).