Even though supercapacitors can source lots of current, they won't just push it into a regulator or other load. Generally, a load draws the current it wants from the source, so as long as your LEDs do not draw more than the regulator can handle, it should be okay.
As for using a switching regulator to charge a supercapacitor, the main issue I was thinking of was the uncharged capacitor looking like a short to the regulator and thus, the regulator not turning on. There could be other issues after that point, but as I mentioned, I do not have much experience with supercapacitors. You can probably find plenty of advice elsewhere online though. For the D24V150F12 regulator specifically, outputting 8A with 40V is at the very edge of its capabilities, so it is not surprising to me that it would run very hot or even overheat.