The mod page says Curious Lanterns is compatible with Hardcore Torches, but it appears that in the 1.20.1 version of the mod, Hardcore Torches lanterns do not get rendered on the player model. I tried to look at the code, and I think it might be as simple as someone forgetting to add them to this list in the main mod file:

I might not understand how the mod works, but I would suggest looking into this list as a possible reason for the bug. The ids that would need to be added are:

hardcore_torches:unlit_lantern hardcore_torches:lit_lantern hardcore_torches:unlit_soul_lantern hardcore_torches:lit_soul_lantern

apologies for not getting to this until just now.
it seems i neglected to add compatibility to versions other than the 1.18.2 version, which is entirely on me.
this should be resolved in the upcoming updates.