I found the cause of the LS-0013 error—it was related to the web browser settings. The first time I installed the Epic Games Launcher and Fortnite, I did it through Google Chrome. Later, I changed my default web browser to Microsoft Edge in the system settings for a dumb reason.
Even though I could log in to the Epic Games Launcher, UEFN, Unreal Engine 5.5.4, and all Epic Games sites like the Creator Portal without issues, my account was still being blocked under webcache_4430 because it was originally linked to Google Chrome. This caused various problems, including Fortnite not being recognized for login.
After switching my default browser back to Chrome in the PC settings, the LS-0013 issue was resolved.