This looks very similar to the UE 5.8.1 → 5.8.2 Launcher issue I investigated.
In my case the first Update had already installed the actual UE 5.8.2 / CL 56702186 Engine files, but the Launcher metadata remained at 5.8.1 and kept offering Update/Resume.
I documented the Launcher logs and added a read-only PowerShell diagnostic plus an optional workaround here:
Epic Games Launcher stuck on Resume after UE 5.8.2 update – manifest commit never completes
I would recommend running only the default DRY RUN first. It does not modify anything and will check whether your physical Engine is already 5.8.2 / CL 56702186 and whether the remaining problem matches the same Launcher manifest/pending-state issue.