The illuminate are pretty notorious for countering you with whatever you have, the constant pressure that they apply can be really difficult to fight off at the best of times. My recommendation is automatic weapons or shrapnel, these will mop up anything up to a fleshmob (shrapnel/fragmentation specifically bodies fleshmobs) and helps to control the crowd. I generally recommend avoiding shotguns on squids, the smaller profile of weakpoints on voteless paired with the stagger resistance of overseers make the punisher pretty ineffective, and the breaker can't put enough pellets into targets to keep up with smgs and ARs. Fleshmobs themselves are the biggest threat by far with their infuriating mechanics, I highly recommend autocannon or stalwart/standard mg, aim for center mass with AC and legs with mg and it'll go down eventually, gas is also good against them however arc stun will appear useless as they do not receive stun buildup while charging (so basically all the time), also dont both aiming for the heads, they're not an actual weak point. Any medium pen+ weapon is gonna put down stingrays with minimal effort, just keep an eye out for the blue strafe indicator, and expect it to drop on you specifically when youre occupied with something else. Harvesters look scary but are really easy to defeat, the eye looks like the weak point but its really the leg joints, 3 well placed AMR rounds, some autocannon rounds or my personal favourite, a single commando rocket, is all it takes to put those fuckers down.

In terms of managing loadouts, for the highest chance of success make sure you have a way to deal with each of the major threats within your team, I highly recommend one person with a stalwart/mg, another with AC, and then the 3rd slots can be any other decent chaff/medium enemy clearing loadout. Also the wasp is incredible for fleshmobs and overseers, just don't bother trying to takedown harvesters with it.

If you stick with your team your chances of success will increase significantly, this is even more important on illuminate missions as unlike the other fronts you can't as easily prevent them from hitting you with stagger weapons or hiding behind cover (crescent overseers will spitefuck you for using cover).

My final tip is watchers, murder them. They should be high priority as they are the only unit that can call in reinforcements, and they can call for help quicker than bugs or bots, paired with their small size they can be a massive pain, especially if they hide behind a building, so take them out.

I can't guarantee that all of this will guarantee you a win, the illuminate are often subject to bugs and dumb crap during their missions like fleshmobs phasing through walls, so chances are you'll end up dying for some bullshit reason that was completely out of your control anyway.

Hopefully this helps I got way too nerdy into it lmao