TL;DR: Search your \Windows folder for api-ms-win-downlevel-kernel. it should be found in System32\CompatTel or in one of possibly many \Windows\winsxs\amd64_microsoft-windows-xx folders. Copying the file to System32 should resolve the issue.
Long Version: I had a similar issue after installing PSO2 through the Epic Launcher. Starting up the launcher threw an windows error message bx that said api-ms-win-downlevel-kernel32-l2-1-0.dll was missing. Yet, the launcher still started up, although trying to launch PSO2 gave the LS-0013 error and Epic Launcher told me to verify the files/download (did that twice to no avail).
I was able to find api-ms-win-downlevel-kernel32-l2-1-0.dll in System32\CompatTel (CompatTel is for Microsoft Compatibility Telemetry service, CompatTelRunner.exe, and part of the Windows Update KB 2976978; there is also a api-ms-win-downlevel-kernel32-l1-1-0.dll in there). I copied l1 and l2 to both my Windows\System32 folder (there was no prompt to overwrite) as well as to the Win32 and Win64 folders in my Epic Launcher install location (Program Files (x86)\Epic Games\Launcher\Engine\Binaries) where again there was no prompt to overwrite.
I'm not sure if having the dll filename all lowercase matters or not, but I did it anyway.
Also, I killed the process tree of Epic Launcher through the Task Manager's Processes tab and restart the Launcher to check if the issue was fixed. It's good to go.
Anyway, I think it's a little strange that Epic Launcher required this dll file, but for whatever reason left them out of the install package. Other api-ms-win-xx.dll files are included in the Epic Launcher Binaries folder. I can't remember if I ran EpicInstaller-10.xx.msi as administrator or not. That could have had something to do with it(?)