Thanks, namida, that was indeed the issue!
I had set up L3DEdit in a separate folder to L3D - partly because I downloaded L3DEdit first, hoping to maybe be able to extract the graphics even without the game itself, and partly because I wanted to keep the original levels and the editor separate as long as I don't intend to make any actual L3D levels.
There was one line in the manual that had me suspect L3DEdit would have to go in the same folder as the game itself, but I wasn't quite sure:
L3DEdit will find the other one by itself as long as it’s in the same folder – which it should be if you’re using actual L3D data files
I wasn't sure whether this "should" was an imperative and referring to L3DEdit (i.e. "L3DEdit should be in the same folder"), or whether it was merely describing likelihood and referring to the BLK and level files (i.e. "if you're working with the original L3D levels, those should naturally be in the same folder as a result of the installation").
Regarding the .ini files, I have both L3DEditPresets.ini and L3DUtils.ini in the same folder now, together with everything else.