In seconds? Wow. I think you’re right, you might need more than a small fan!
It might be worth exploring heat pipes or peltier effect coolers. The latter makes the problem worse (they are inefficient and generate a lot of heat) but your LED can be locally cooler if you can e.g. move all that extra heat into a big heatsink (also condensation can be problematic).
One cheap source of heat pipes for testing could be old graphics cards – they often outperform simple copper heat sinks. Use thermal epoxy to stick your LED to it and see if the performance is acceptable. On the exotic end of things, you could also water/oil cool it, or (carefully) make your own thermal grease from industrial diamond powder for a small boost in thermal conductivity.
Also even at 95% efficiency, it sounds like your boost converter has some heat to dump too!