Against huge swarms of strike craft, Vasari will need to combine multiple units.  Your flak frigates will rip apart fighters, while your own fighters will rip apart enemy bombers; these are the offensive side of your counter.  On the defensive side, you need Kortul battleship and a Skirantra carrier.  The Kortul can jam weapons, disabling enemy strike craft nearby.  The Skirantra can use its "repair cloud" ability to heal everything nearby.  This is the only ability in the game that can repair strike craft.

Now, defeating guardians is another matter entirely.  Skirmishers are your best bet here, although not entirely for damage.  They have an ability that will cause guardians (and carriers for that matter) to deplete their antimatter quickly.  Combine this with the fact that skirmishers are at least the toughest of light frigates and you have a solid combo.

Heavy cruisers are killed by bombers easily, but if you can use your defensive abilities (including overseers for healing) they can prove surprisingly resilient.  As I've discovered as of the new patch (which buffed reintegration) you can actually beat carriers equipped with bombers by using enforcers.  It seems that if there aren't enough bombers around, the enforcers won't die very quickly at all and can kill the carriers first.  That said, if the bombers are in significant number, reintegration won't help you and then the bombers start to rule.

You'd still want to use flaks and fighters as your primary counter, as well as hoshikos for healing purposes.  Aside from that, the Kol battleship with flak burst is great for clearing large numbers of strike craft.  Dunov with magnetize is also useful, but won't stop huge swarms.

Cobalts with saboutage reactor can also be useful against guardians and carriers.

If you are worried about Guardians, try using Subverters. Their distortion field (aoe disable) will be really useful if they are all bunched up (activate its abilities manually for best effect though since the autocast isn't very smart about it) and it has a longer range than repulse. Skirmishers with the interference will also keep those guardians at bay and it will also make the carriers have to spend more antimatter to reproduce strikecraft(can anyone confirm?). Having a Kortul with Jam weapons is a must.

If they are massing carriers and guardians, their only significant source of dmg comes from the strikecraft so make sure you get as many strikecraft jammed with the Kortul as possible. With a lot of flak, skirmishers, some fighters of your own and a couple subverters and maybe some overseers for heal if you can afford it, you should do fine. Of note, flak automatically targets strikecraft and will often wander off on its own to target them. However, Sentinels have multiple banks it can fire from and when you are using them to attack the enemy fleet, they will still auto-fire against strikecraft within their range.

You are going to lose a few ships, that is pretty inevitable due to their large number of strikecraft. What you need to do is make sure you can pick off some of their ships before they run out of the gravity well to rebuild their strikecraft and heal (though you could chase them down too). Your subverters will come in handy here as well as your Gravity Warhead on your death egg.

Returning Armada on the Phase Gates will warp in ships for you at the planet the phase gate is at. It is random mix but is much cheaper and faster than building the ships from the factory. I do not believe it will give you any subverters or overseers though.

As for TEC, flak burst is very very good against it. Two Kols with lvl 3 flak burst will devastate strikecraft. EMP Charge on the Dunov is also a addition to the Kols for combating guardians and carriers as will dmg their shields and deplete their antimatter.

Consider yourself lucky.  There was a point in time with this game where massed carriers of any race REALLY ruled, and in my opinion (which is disputed by some, I guess)  it was uncounterable except for massed carriers of your own.  The so-called "counters" to the carrier (light frigs and flaks) were at that time useless (in my humble opinion - others here will differ).  Even worse, carriers were the fastest ship around, next to scouts, so they could REALLY kite you.  There have been nerfs and adjustments since then, and now things are a little different I hear.

If all he literally has is drone hosts and shield guardians, then ON PAPER your perfect counter is the skirmisher.  I know you said you tried them, but are you absolutely sure you built enough of them?  ON PAPER you should cream him with those, but note that you will need to build a TON of them - several per cruiser you need to take out (3 or 4 at bare MINIMUM I guess).  You will need a TON, but note that you will still be spending less money to take out his army than what he spent to deploy it.  On paper, that is your counter, and also research an ability for your skirmishers called "interference," and if you can, "reintegration."  This is a rough number off the top of my head, but if he has 20 drone hosts and 10 shield guardians, I'd say you're talking 90 skirmishers minimum (at least back in the day).  Did you deploy that many?  A 1 to 1 ratio isn't going to cut it here.

Now, that's all ON PAPER.  But I can tell you that what is "on paper" didn't do jack crap back in the day before carriers were nerfed and adjustments were made.  But this is the first thing I'd try right now, if I were you.  Just spam Skirmishers.

A possible wrench thrown in your works:  If he has a Progenitor Mothership sitting with his carriers and guardians, or even worse - two of them - you are screwed, and this simple strategy just became a hell of a lot more complicated than what can be described here.  Suffice to say, you will need to take down those Progens with something that drains antimatter (a Kortul perhaps?).  Good luck trying.

Other notes:

You said you tried heavy cruisers.  Don't use those if he has bombers in the grav well (and apparently he does), as they are the hard counter to your heavy cruisers.  In other words, your heavy cruisers aren't countering jack crap in that situation, rather they are being countered themselves.

Flak used to not be effective back in the day.  It might be now, but note that if it is, it will only be effective against fighters, not bombers so much.  Plus, you are really only countering the strikecraft themselves with flak, not the carriers.  You really want to kill the carriers themselves.  If you try it, Flak should be used judiciously and specifically and with purpose - say jumping in with a ton of those first, letting them stay in the grav well for a while killing all the strikecraft (IT WILL TAKE A WHILE), and only then jumping in with a fleet to deal with all the cruisers.

I know you said you didn't want to use your own carriers against his carriers (I didn't either back in the day - I'd take a loss, just to try something different).  And I know you said you tried carriers but he rebuilt strikecraft too fast.  This simply means that you didn't deploy enough carriers.  If you deploy an equal number of resources in carriers to his, and set them to fighers, you WILL either win or draw against him, but not lose.  Note that the vasari carrier is cheaper than his, and has less strikecraft, so you will need to deploy MORE carriers than him, but not necessarily more resources in carriers than him, comprehende'?

Lastly, pick a Kortul as your opening capship, and nurse it up in levels, getting the "jam weapons" ability for it whenever you can.

Keep it simple first, though.  Just spam a crapload of skirmishers (A CRAPLOAD), and buy those special abilities for them if you can.  Report back here with your results.

Also when being the TEC, build a couple marzas with the missile barrage ability + you pretty much just own everything. You just need some of those frigs or kols mentioned above to cover them from strikecraft

You peopel never learn. Since you say late game. Go for Kostura. and kortul egg combo. PS: By the way does the gravity warhead affect squadrons? Because if it does it makes them slow and easy prey. And make sure you mass those skirmishers and upgrade weapons.