To me, this sounds like the basic observation here is that when running as admin, there is no SSL error, but when running normal user, there is an SSL error. That does actually tie in with the difference between running the Launcher exe yourself which you marked as must be ran as admin, and running the uproject file, which would very likely not respect that setting.
When you run the uproject file, a program called UnrealVersionSelector is ran. This will in turn run the appropriate editor exe, which in turn always runs the Launcher exe I believe. None of this is probably done as admin, which would explain the odd difference between those paths.
The only explanation I can think of for now would be that somehow your user account does not trust our certificate, or one in our key chain, but the admin account does. Does that ring any bells with you? I guess on a similar thought, per-user internet options could be different between your normal user account and the admin one.