I tested the latest version. Now I can browse the menu(there are minor glitches, but I am not sure whether it is the fault of the game or drivers: https://my.mixtape.moe/udiwkh.png), the background music plays but no success when trying to play a level.
Here I uploaded the output of Import.exe: https://pastebin.com/8FJwRgyG
The "missing" tilesets are where they should be anyway: https://my.mixtape.moe/cojcog.png

The error logs are about Animations/Jazz/spring.png.res and other "spring-related" files related to the other characters. The files themselves have filenames with capital "S".
After manually changing these, it complains about Animations/Pickup/fast_fire_lori.png.res
This file is missing.
Is it possible that using LK Avalon version of JJ2 1.24 breaks it? I don't have the original English version, and on 1.23 there are even more absent files.