Expression must have a class type, creating a class reference

Yes, I was able to reproduce the error. Sorry for the delay in responding. As a work around you could make a function to get the gamemode when needed. Something like this (this is assuming you have declared the gamemode object in your header file)

In your .h file

ABrawlerGameMode* GetGameMode();

in your .cpp file

ABrawlerGameMode* ABrawler_Character_Player::GetGameMode()
{
    if (GameMode != NULL) return GameMode;
    else
    {
       GameMode = Cast<ABrawlerGameMode>(UGameplayStatics::GetGameMode(GetWorld()));
       return GameMode;
    }
}

So your call in SomeFunction() would be something like this

BMovementComponent->AddInputVector(FVector(0, 0, -1) * GetGameMode()->Gravity, true)

Hope this helps. It seems that you cant define it in the constructor since GameWorld() is null. You might want to try defining the GameMode variable in PostInitializeComponents().