The gamemode isn’t a good place for that. Getting a reference to an actor without searching the entire lists of actors is also not going to happen just by asking the level.
This is a common situation where you usually want to register actors into a certain system such as a GameInstanceSubsystem. Programming Subsystems | Unreal Engine 4.27 Documentation
You could for example make an CharacterSpawnerSubsystem which spawns and keeps track of characters, then simply request those characters when you need them. Technically you can also make that spawner just an actor or object but see which fits your needs best. For me it is usually the subsystem.