Doesn't Photoshop do everything in L a b ? Why would they have a setting for Perceptual if it doesn't actually do it from PP to aRGB. You would think that would be explained somewhere in settings. I suppose that aRGB to sRGB actually does Perceptual and the other three.
Again, you are failing to understand the fundamental ICC color transforms...(and no, Photoshop switched to CIE XYZ several-many versions ago).
Photoshop (and Adobe in general) has done a disservice to the industry by allowing people to select what are actually, unavailable rendering intents since, well, forever. I don't know why and many of us have complained since, well Photoshop 6, 7 CS, +. To no avail.
And no...ARGB to sRGB won't supply perceptual unless the sRGB has a perceptual rendering intent–which is ONLY available using one of the special sRGB perceptual V4 profiles.
Look bud, you are swimming upstream without a map. You don't know what you don't know. If you want to learn, cool...but what you've "assumed" so far is off base and screwed up.
Go back to the basics...ICC color management starts with an input profile (scanner or camera) which is transformed into a working space (PP RGB, ARGB, sRGB or whatever). You work on the images...if you want to go out to another RGB color space (like the web) you are limited to matrix color space transforms which if you don't have a v4 RGB space (like the V4 sRGB perceptual) means RelCol. If you go to from a "working space" to "output profile" you'll have all 4 of the rendering intents and all 4 will work (to the extent that the profile maker has done their homework).
Look, for the purposes of simplicity, forget about those exotic alternative RGB color spaces. If you are using ACR or LR, use ProPhoto RGB in 16 bit. Once you have that open as your working space, depending on where you want the image to be output to, pick your profile–if you want a perceptual sRGB (for the web), use the sRGB V4 profile. If you are going out to a printer (table based profiles), use the printer profile to soft proof the image and select the best rendering intent for THAT image...
You are making this more complicated that it should be and you lack some basic understanding of what the ICC color management workflow is intended to be.