@ Ekko, who's thoughtful responses I appreciate...
I think you and others are exagerrating the difficulty in getting three military labs, so let me describe my typical start as Advent that has lead me to my opinion, and we can go from there...
I scout planets touching my HW to get a sense of what I need to do for colonization and labs...I also start with a progenitor...the following outline is general, and is prone to changes depending on the situation...I just use it to show how I normally get my 3 mil labs and 2 civ labs...
1. If HW connects to only one asteroid/moon and only other viable nearby colonizables are ice/volcanic, then I go one mil lab and one civ lab on my HW...after progenitor is complete, I scuttle cap ship factory and build 2nd civ lab...the 1 mil lab is for corvettes, which I use to aid in expansion (deal with krosovs and LRMs)...once 2nd civ lab is built, I get appropriate colonization tech...my 2nd mil lab goes on asteroid/moon, and 3rd mil lab goes on the newly colonized ice/volcanic...I have 3 mil labs and 2 civ labs, with no logistic upgrades...
2. If HW connects to only one asteroid/moon but there is another viable nearby colonizable that isn't ice/volcanic (like a desert/terran with reasonable militia), then I go 2 mil labs on HW...after progenitor is complete, I scuttle cap ship factory and build 1 civ lab...2nd civ lab goes on asteroid/moon, 3rd mil lab goes on newly colonized desert/terran...I have 3 mil labs and 2 civ labs, with no logistic upgrades...
3. If HW connects to multiple asteroids/moons, then I go 2 mil labs on HW...after progenitor is complete, I scuttle cap ship factory...I will build a 3rd lab on HW, but it depends on what is nearby...if I find a nice string of asteroids, moons, or terran/deserts with reasonable militias leading towards my enemy, I may just ignore the ice/volcanic and not even bother with civ labs on HW...usually though, my 3rd lab on HW is civ, and I'll put 1 civ lab on one asteroid/moon and 1 mil lab on the other asteroid/moon...even if I don't need the colonization techs, getting culture ain't a bad economic boost to have when I get the chance...I have 3 mil labs and 2 civ labs, with no logistic upgrades...
There are some other guidelines I follow...if I have a choice, I try to put military labs on rear worlds so I have access to my prototypes even if I lose planets...also, if for whatever reason I find myself 3 jumps or less from an enemy, I may avoid labs on that frontline planet if I can, but that's more to get factories and repair bays up faster, not necessarily because of logistics...
You make a valid point that extra labs may affect your ability to get multiple factories, however I don't find that to be much of an issue in most games...in the case of an asteroid/moon, you must get the logistic upgrade in order to have 2 factories, regardless of whether you build a lab there or not...in the case of a terran/desert, it doesn't matter since they have at least 3 slots to begin with...it's only with ice/volcanics where it might be an issue, and even then two conditions would need to be met before it'd be a real problem: 1) the ice/volcanic is the frontline world and 2) you can't get 2 asteroids/moons...it happens, but it's not likely...
Anyway, from my experience the extra lab really does not increase the need for logistic upgrades in most situations...I may delay my 3rd lab in favor of getting factories/repair bays up faster, but asteroids/moons always need that upgrade to get 2 factories independent of what else you put there (or don't put there)...terran/deserts aren't an issue since they start with 3 slots, and ice/volcanics are only potential problems if you can't get 2 asteroids/moons...
Basically, from my strategy illuminators are only costing me an extra lab, and the prototype tech is a little more expensive than a tier 2 (like LRM)...no need for extra logistic upgrades except in extreme situations (and those would be the situations to not try and get illuminators)...
One possible issue with my strategy is that you must upgrade the logistics of the HW in order to build a culture center there, but by the time I'm actually ready to get culture I can afford the logistic upgrade no problem...plus, I usually use that 6th slot on the HW for another factory or another mil lab, depending on how things are going...so getting that upgrade again would happen regardless of whether I rush 2 or rush 3 mil labs...
That's how I do things...not saying it's perfect, but it works for me...spamming illuminators after your initial corvette/LF/flak skirmishing is not a "default goto" strategy, but I always like having the option...you definitely see LFs in large numbers quite frequently because of the common flak build ups, and Illuminators are great to have...
It also surprises me that you are trying to use carriers so early in the game...what are you building them for exactly? Early game has tons of flak, corvettes, and LFs, all of which heavily counter carriers in some manner...furthermore, the build time on the carrier + the build time on the squadron makes it a rather rough unit to try and use early game, it just takes too long to get your firepower in the fast pace rushing that goes on...I won't say they don't have their place (good for dealing with Vasari SB rushes on HW I suppose) but I just can't see myself prefering those over illuminators that early in the game...