How do I get Unreal Engine registered in the Epic Games Launcher?

Welcome Joe.

im not entirely sure but take a look at this registry keys


[HKEY_CURRENT_USER\Software\Epic Games\Unreal Engine\Builds]
"4.27"="C:/ue/OUE4.27"

keep in mind that the engine version in the uproject needs to match one of the key names.
sometimes you get a uuid like {E3193045-4999-5EBB-E26A-XXXXXXXXXXXXXXX}
these tend to be unique per machine. and become a pain if the project is shared among users or computers.