I'm not an expert, I fought one C3, since 2014. But I won and can report on that. You can maybe can derive something from my experience.
I started the war because I couldn't remember anything about war, and wanted a refresher to guide my thinking on how to set up my defenses. I didn't intend/think I would win it. the country had a defense index of 121.
I tried things like luring in the airforce buy attacking defenses with units that had tons of DefAA. I tried attacking the airforce with attack airwings it was all naive poking around not knowing what do and trying to remember what to do. Lost my entire off airforce right away. (I'm interested modern tactics for dealing with airforce, but don't expect anyone is willing to share here.)
In the end one thing I rediscovered/remembered was that there is no need to bother with the air force in conquering a C3. Just ignore them and whistle in the dark.
First thing I remembered is that C3 army will not attack you if you don't attack it (borg philosophy, maybe). So I set random units driving all over it, they would stop moving whenever they encountered "weapons"(if anyone knows what weapons cause this I'm interested. It didn't seem to be all weapon.) I was able to paint 95% of the country and discovered where all the major weapons were by just driving around. (I'd imagine in a real war with a human, they would send something to stop you from doing this.) This dropped the war index a good bit.
Next, I remembered/accidentally rediscovered that the air force will not respond to ground attacks on targets. By this point I only had several hundred of each ground offense weapon, but plenty ammo. (plenty of ground attack ammo is key ~50k). But I knew there were plenty of targets that were not defended. So I sent my ragtag collection of ground assault units accompanied by whatever defensive units were left on a cross country tour popping off any undefended targets they came upon. (get at least 3 long range radar planes to locate undefended targets.) Didn't have any losses except for when I got to close to the enemies mobile ground units. So just stay away from those, if you attack even undefended targets near them, they will retaliate.
In summary my takeaway on beating a C3 is just assemble maybe like 500 total ground weapons (armored vehicles/artillery/ etc. with around 50,000 rounds of matching ammo. I think one of the ground weapons doesn't even use ammo, so if could be fun to see if just using those would work. I also had some defensive units carrying whatever several hundred "def" missile types I had left. I sent them along for good measure, but it may not have been necessary. Then just sneak across the border, hide from the enemies ground defenses, and terrorize whatever undefended targets you find.
Perhaps I got lucky, I don't know, but that's the story of my C3 win.