If it's causing you major grief, use a Radiance with detonate antimatter. That will shut down power surge (at least until disruptive strikes shuts you down, so ensure you have the firepower to make it stick). The one risk is that the Radiance is absolutely the wrong choice of capital ship if your opponent is leading Vulkoras or Skirantra, so unless you're psychic this will have to be your second capital ship.
I could have quite a large fleet (like 10+ Illuminators and my own capital ship)
That is not a large fleet at all; that's actually a very small fleet, and I wouldn't engage a high combat-value capital ship like the Kortul with such a small force. High combat value capital ships are risky to engage with small fleets like this, and the Kortul is perhaps the most deadly of all.
Your only real option is to lead with Halcyon against Vasari. Focus on his frigates and try to isolate the Kortul. You can take it down if your fleet is large enough and it doesn't have support, but don't even try if it's got backup. I only attack capital ships when I know I can make it count.
I couldn't take out his defense. He gets a heap of Flak frigates and wins. I think I made some mistakes with micro here, letting my fighters and bombers attack his Kortul instead of his buildings, but I'd anticipated the badly-hurt Kortul would have to flee
He probably had a repair platform here. A single overseer would also make your attempts to take down the Kortul futile.
Now, against massed flak, you want to be very careful with strike craft. You should probably be phasing in some disciples or destras. Vasari sentinels are particularly nasty and you need to hit them with the right units. You should probably try to stay one step ahead of the enemy; just presume he's going to counter your carriers so your own counter-measures come on the field in time to beat his.
Micro is particularly critical with strike craft.
The big problem is that the Advent illuminator just isn't very effective against capital ships. Vasari and TEC can mass up a chunk of LRF to beat back the Kortul with sheer firepower, but since the illuminator bug fix the illum really can't manage that well. This leaves you with very few options, and this is affecting Advent's ability to counter more than just the Kortul.
Strangely enough nobody seems to have mentioned this tactic anywhere, so I don't seem able to just read other topics
I love the Kortul opener, it's a great strategy, but the bottom line is that it's still a second tier strategy compared to the carrier opener. Still very good if you know what your doing, but not a top-pick strategy. I personally love it, the player you fought clearly did as well, but it's really more along-side the Vulkoras rush as something that's most powerful if your opponent doesn't see it coming or doesn't know how to counter it.