Damage via weapons on capitals cannot be balanced.  They are ability, not DPS ships.  In the strictest sense, they are support ships.  The ONLY time the Kol becomes DPS efficient is if you've got a Vasari buddy who gives you an Armament Pact and you've already got Finest Hour.  At that point though, the Kol is basically a glorified frigate which no capital should be and you're into the late-game in which titans are going to be playing a much larger role than caps with few exceptions (Progenitors and Jarrasuls for instance).

The problems with the Kol are that...

1. It is an AM hog.  Three abilities all burning through AM, two of which don't help your fleet at all.  Flak Burst is good late-game when you already have a lot of flak because it can really help in that regard, but odds are you won't have the AM because of GRG and AFF.
2. GRG is worthless in fleet battles because you have focused fire and focused fire brings up mitigation.  Mitigation makes raw damage from GRG useless.  Why is Snipe good?  Because you can engage from a stupendous distance and unload on them before they even get a chance to fire back.  Oh, and they are still at low mitigation when you hit them first because Snipe has such a long range.  Mitigation is the bane of raw DPS and is why GRG against anything other than structures is worthless (bear in mind that pirates have a stupendously powerful armor type which renders capital abilities used against them only 50% as useful as default plus their armor itself) and structures generally have very high armor.  Oh, and the Vasari can phase out their structures and the Advent can shield theirs, so really, it's only useful against TEC structures.  But seeing as how you are TEC and want to blow up structures, why don't you just bring Orgovs?
3. AFF does nothing to help your fleet.  Yes, it keeps the Kol alive longer.  So?  Beyond that, critical mass bombers will kill the Kol in maybe two passes instead.  Recall that the Akkan has almost as much health as the Kol and it is far better as a fleet ship.  Also remember that the Corsev is the best tank in the game, especially against Advent which have no ability to focus-fire (Unity Mass notwithstanding).  The more of its allies you kill, the more it heals.  There's a reason I nerfed the Corsev hard in Rebalanced Races.  Back to AFF though, you might say, but FB kills bombers!  No it doesn't.  FB drains a ton of AM and does positively NOTHING to stop the alpha-strike of bombers.  You'd have to have multiple Kols to do that which is a huge cost investment and it means fewer of more useful capitals.  You build three Kols and have to obsessively micro them to get any decent sort of SC stopping happening.  I build 42 Gardas, slightly less fleet supply than you used.  I'll have more even SC stopping power spread throughout my fleet which can't be as easily micro'd around (FB's range is short, so the window in which to strike is very short, making microing multiple Kols a pain against bomber waves).
4. FB is handicapped because it is hard to hit with, doesn't disrupt bomber passes (which both Jam Weapons and TKP do), and costs a ton of AM on an already AM hungry ship.

Finest Hour is something that people seem to think will make everything better.  The thing is, it doesn't.  late-game FF will bring down the Kol one wan or another and as mentioned before, not much slower than the more viable Akkan and far slower than the Corsev.

Kol needs love and unless facing an AI or hordes of bombers (at which point, I recommend building 3), I don't recommend people building it.  In my eyes, I do not see any time when you would want 1 or 2 of them unless you already had some special things set up such as an L6 Dunov with Flux Field or an Armament Pact (which really only works that well if the Kol itself hits L6).

The Kol looks cool but is simply not practical.

Making Kol effective is simply a matter of ability reworking. Core values of the unit itself are fine.

Flak Burst: currently offers no cover from SC to disrupt the alpha-strike of bomber swarms. Both JamWep & TKPush completely can negate the initial damage runs of a huge bomber swarm prolonging the useful-ness of capships immensely. I believe Flak Burst should be changed to a duration effect with damage over time as the SC fly through the AOE and a debuff to accuracy for a period of time after being hit by the flak. Perhaps figures as follows:

AM Cost: 75-85-95-95

Cooldown: 20-15-10-10

Duration: 10-13-18-25

Range: 3000-3380-3760-4140

DOT: 8-10-12-15

Accuracy Debuf 20%: 30-40-50-60

This provides cover, damage support and removes the obnoxious micro-intense requirements it currently has for usefulness.

GRG: Actually not terrible once you solve the AM problem but needs a more powerful debuff. Or it needs a secondary AOE effect which is what I advocate. Something like "Projectile splinters on impact damaging nearby objects" and would do 25% of normal damage against 5 targets within 1200 range of original target.

AFF: Effect at each level should be cut in half and made passive.

Easy Peasy Lemon Squeezy. Best of all, none of this requires code breaking changes. All of the needed modding is available. I could mod it myself if I had the time to mod.

A very MANLY brick, to be fair....

Lol....