The whole idea of LA CA (which I am arguing for in post 19; ideally you want 4 Light Cruiser Battery & only 1 Heavy) is to melt enemy screens, which then allows your torpedo boats to fire at enemy capitals. Enemy screens cannot attack them with light attack, since they can only target the closest non-empty group (direct quote from wiki), and if they have 100% screening ratio (which is why I said well screened in post 26) torpedos cant touch the CA either.
They arent really designed to fight HA CA (or BB/BC for that matter) head on, without support from your other capitals. Reason being that heavy guns can attack the second layer, and assuming 1944 CA with armour IV you get 25 armour, which is pierced by at least Medium Battery II (27.5 piercing), they will deal a lot of damage to LA CA with or without armour. Which is the reason why I suggest no armour on it because they are not going to be as useful, but cost you in terms of IC and speed, which is crucial to evasion. So on that note I agree that LA CA with get mostly battered by your beefer CA on their own, but hopefully the enemy screens are already severely weakened before that and torpedos can do significant damage to enemy capitals already.
But this scenario is quite rare as well. If you have with any kind of capital ships given to you from the start you would doomstack them together, and those crappy capitals are bound to have higher visibility which means they are being mostly targeted first. So the beefer CA will target the crappy capitals first, which again support the idea that no armour is really needed in the LA CA.
If you are going to use them for patrol, then yea the LA CA will get sunk by HA CA (or your beefer CA for that matter). But the direct response would be to use LA CL because heavy attack is much less effective against screens compared to light attack. And the direct response against LA CL would be, well, LA CA. And the merry go round goes on and on.