If TEC are three jumps away, get a starbase operation on your homeworld. Advent is very much in the same position as 1.181 Vasari; at close range, you have no hope of winning a frigate battle.
it is possible to get a starbase up and running if you have sovas sucking your credits away, though you are fighting a losing battle that will inevitably result in needing lots of feed, running, or needing support. Since the TEC are getting all the income they are sucking away, you have NO chance unless by some miracle you have a hidden neutral stash that the TEC don't know about and that a vasari player hasn't fought you for (which is practically NEVER! lol).
When facing equally skilled players this early match up can be the deciding factor and push your team to lose. Sovas just rape advent, since advent don't have any economic advantage early on. spamming disciples does you no good (yes even with their tech level 3 upgrade), since the TEC can and will spam lrms to support their cobalts. Since it is far easier for TEC to get their lrms, the advent player already lacking in economy cannot react to this, especially since going defensive early means that the starting roid is up for grabs (so bye bye logistics slots, HELLO incoming enemy fleet spam on HW). Also, spamming scouts won't work effectively since economically you are at a large disadvantage. You won't have enough Illuminators up to really deal with the enemy, and since illums suck at focus fire anyways you won't scare those sova's away.
and we are talking about a close quarters TEC player still.
Oh and Greyfox good post!
I would add to your post that advent don't have an early economic advantage at all when compared to the other races. The progenitor ability doesn't do enough really, and in a lot of games I find that if I don't go Halcyon i don't have a chance!
This could help to rebalance advent. As to what that economic advantage would be when compared to the other races economic advantages (sova's, low cost economic tech for TEC, Vasari upgrade efficiency and tech 1 LRM, LRM cost reduction via slavery to make kanrak spam cheaper than illum spam, and easy neutral control), I have no idea.
Edit: Here is an idea! (also a possible answer to your post below Darvin3)
Another way to go about dealing with this economic problem is to have advent Fighter/bomber structure cost about half of what it does currently, either through a tech lvl 1 defense upgrade, or have it inherent. This would really go a LONG way to making any rush against advent in the early game much more difficult to pull off. Also, reduce its tactical slot requirement if needed(again either inherently or via the upgrade). It also makes sense for advent to have a cheaper fighter defense structure because that's what they do best, fighters and defense. Advent are Defensive in a lot of ways, the enemy shouldn't be able to get a foothold so easily on your HW. This would counter sova/LF spam and give you killing power while you slowly build up your fleet. This could also counter 2 vasari carriers with a starbase on your HW (once scramble bombers is fixed). Also, you could reasonably defend your asteroid with these improved fighter defenses, though it would still be much weaker than your HW.
This is not overpowered, because you wouldn't be able to then ADVANCE your fighter defenses to be used against your enemy. You would have just made yourself a heck of a lot harder to hurt. This wouldn't be overpowered in the end game because the enemy always has Kostura's to get rid of a heavily fortified advent base. As it stands, Advent should be the hardest to rush because they are such a defensive race.