Yes I expect it will cloud liquid soap, although I've not tried lanolin in liquid soap (I have in bar soap). I can't say whether lanolin would cause the LS to be just translucent or fully opaque -- that depends on the amount of lanolin used and the rest of the recipe.
I have never once bothered with checking for cloudiness, even when I expect the soap will be transparent. I have enough experience making liquid soap enough to have confidence in my technique and my recipes, and I don't go overboard on the superfat. I zap test to check that the alkali is fully reacted and that's all I do.
All the "cloud test" tells you is whether or not there is something to cause cloudiness. The cloudiness could certainly be excess unreacted fat, but it could also easily be stearic or palmitic soap or unsaponifiables. A cloudy result doesn't even tell you positively whether the soap is made correctly or not, unless you're absolutely certain of your recipe and your ingredients. And it also doesn't mean that the soap needs to cook longer. If you get a cloudy result, but the zap test tells you there's no excess alkali, then the soap is as done as it's going to get and more cooking won't solve that problem.
If the cloudiness is caused by unreacted fat or fatty acids or unsaponifiable chemicals, the opaque material will eventually separate out of the diluted soap. Skim the opaque layer off, use the rest, and take notes for future batches.
