Carrier bays most emphatically do work.
In the Defend Caspian IV scenario, I had 4 tank cruisers, each with 2 blasters, super tractor, and the rest defenses. 4 Fast missile launching cruisers with 1 carrier bay each, and lots of armor and shields, 6 panthers in the back with 8 MWM each, and to swat fighters I had 3 panthers with a mix of tractors and blasters.
I tethered mixes of fighters rockets and lasers to each tank, and some laser equipped fighters to the middle carrier.
Now, the initial waves of fighters didn’t cause many of my fighters to retreat to their carriers, cause if the attacking fighters attacked a fighter, typically their whole squadron did. But on the other hand, their fighters didn’t last long. I had tried using frigates against the fighters, but the initial wave had such a high priority for the frigates that they just got swamped.
After the first wave, my fighters would start regularly coming in. They chewed up the enemy frigates like they should at max range, often before the frigates came in range of my Fast Missile ships. I saw 2 of my carriers with over 10 fighters waiting for service at one point. It was kind of fun, rooting for my fighters, trailing smoke and flying sloooowly towards my carrier. None of my carriers died with fighters on them. I estimate that I repaired at least 3 full squadrons of fighters on my total of 4 carrier bays.
Armor damage doesn’t trigger the cautious order, only hull damage, so take lots of engine 1, and don’t bother with armor. Fighter engines do lose efficiency, the more engines you have, sadly.
I am going to have to experiment with Tribe Fighters. Their health would pretty much guarantee they survive the first shot, unless they get hit by a tractor beam.