Good rule of thumb to remember. For ASFs expecting to be primarily operating in space, heavy fighters are often better. The rules allow ASFs to use thrusters to rotate. So these things can begin spinning turrets flying in one direction but pointing in different directions. Also, operations in space can mean going up against dropships and so more firepower/armor makes sense.
For atmospheric operations, lights (as interceptors, light fast bombers, zippy recon) and mediums (as more durable dogfighters) are more typically your go-To's.
In atmosphere it's about that ability to turn and burn in a dogfight. Speed and maneuverability count a lot as you are often relying on the ability to get on an opponent's tail and stay there. The mechanics of Battletech mean that ASF fights more closely resemble dogfights from WW2 or Korea than real world modern air combat. In 3025, aside from LRMs, most weapons are direct-fire with no tracking or guidance systems and you want to get on the enemy's six to use them.
Providing close air support falls somewhere in the middle. More tonnage means your ASF can carry more ordnance on hardpoints. It also conceivably means your strafing runs have more punch to them. The scariest thing for enemy ground forces is for you to bring some 100 ton fighter loaded up with bombs. But in atmosphere those slower birds are also going struggle if intercepted.
Another way this commonly gets broken down is by speed profile. 6/9 is regarded a typical dogfighter speed, slower is often the heavies that are attack platforms, faster than that and you potentially have an interceptor (or fast dogfighter, somewhere in-between)
Given the mission setup you've described, I like the Stingray, a medium ASF that can dogfight, strafe, bomb, recon. It's hard to go too far wrong there.
It can attack dropships, (preferably smaller, weaker ones, it's not as potent as a heavy fighter) so it gets a "C" grade on that.
About the only thing it truly can't do is chase down enemy interceptors. If the enemy is flying around in some of the fast 20-ton fighters out there it'll struggle to keep up. But a lot of that can be overcome via tactics such as setting yourself up so the interceptors are headed toward you rather than away from you so you can get a shot off. They melt quickly.