Is there some tech to increase the efficacy of Engineering Sections? For instance, a larger model of the section? I don't like adding several hundred of them to get my ships below 300% AFR. Or is AFR less important than I think? Do I just need to carry sufficient maintenance supplies?
No there are not larger/advanced versions. Engineering space is actually very important in a ship that flies through space for long periods of time, even today's navy ships are mostly 'empty' space for crew to work on the ship (and has been for a looooooooong time). And AFR is less important than you think. What you want to look at is maintenance life.
Also, how do I make morale on my fighters not go down? Is this what Flight Crew Berths are for on my carrier?
Yes.