When the team at the top decides what planes they are going to fly on what routes I'd be amazed if they considered the overtime payments going to the 4 pilots.
If they feel they can fill 500 seats to a destination at a particular time of day then the 380 will fly it. They won't put a 250 seat aircraft on the route just to save a few thousand dollars.
Equally they won't stuff 500 seats into a market that can't support it or desires frequency rather than pure capacity (Asia) just to avoid overtime.
I think there are much bigger issues which determine which aircraft will fly where, than overtime. I doubt we are as important as we might think we are.
Regarding who gets allocated to which aircraft, in the past they have allocated new entrants on the same day based on how many hours they had on starting. I've seen them start crew a day early or late just so as to have them allocated to the aircraft they want if the hours wouldn't have achieved the desired outcome.