After a couple of hours searching for a reflow oven or station in Trondheim's hacker community and several hours of trying to solder the Lumileds LED to the PCB, I hereby declare the component too impractical for this project.
NTNU at Kalvskinnet (or HiST as they were called when I studied there) was kind enough to let me borrow their reflow oven. Thank you very much! Shame it was all to no avail.
Here's the problem child through the lenses of an old Russian microscope.
Thing is, power LEDs get hot. As a rule of thumb, only 30% of the energy driven into a LED will be converted to light, which means that 70% of it will be converted to heat. For our 3W LED, about 2W of heat needs to be dissipated while keeping the junction temperature of the LED below the rated 85 degrees C max. The thermal pad underneath the LED with an incredibly low thermal resistance helps it accomplish this, as long as the heat has somewhere else to easily dissipate. Large copper areas and heat sinks are of great help here.
The application note of the Cree LEDs describes using thermal vias on the footprint to lower the thermal resistance to the large ground plane underneath. There might be a danger of doing so, though, because the solder might be drawn from the pad and down into the via during soldering, which can leave air gaps between the pad and the component. The application note states that this problem is avoided as long as the vias are kept to 0.3 mm in drill diameter or less.
The Lys Bright PCB was designed with 0.3 mm vias, but as I later discovered, OSH Park only supports via drill sizes down to 0.33 mm. I guess they adjusted the size up to 0.33 mm. It's not much of a difference, but enough for solder to start drawing into the vias. Not only that, I found out that the vias weren't connected to the ground plane on the back side at all, so when I tried bridging the vias to the ground plane with large amounts of solder, it went through the vias and pushed on the LED on the other side so that the anode and cathode lost contact with their respective pads.
Too much of an hassle, and requires reflow equipment. That's why I'm moving to larger, hand solder friendly LEDs, and another way of connecting the thermal pad to the PCB.