Here's my opinion on the various documents you mentioned:
LASORS - A well-written ANO, specific for flight crew licensing. LASORS itself does not have any legal value, and there's no equivalent of it in other countries, but it gives a readable overview of what's also in the ANO, but in far more readable format. It contains a number of sections that are not relevant to you at all (CPL, IR, FI, ...) but if you ever wanted to know how to renew your PPL after a seven-year gap (not an uncommon question on here), LASORS has your answer. But for now, I'd just take a look at the structure of LASORS itself, plus a good look at the PPL section and the SEP class rating section, and leave it at that.
AIP - I found that this document was not covered in enough depth in Air Law, or in Flight Preparation for that matter. It's a document which every country produces, and every country uses the same ICAO template. When you go to fly in a foreign country, it's the definitive document on what's different in that country, with the exception of the legal text, which of course may be in a foreign language and thus unreadable. It's well worth having a read of the GEN and ENR sections, but better yet download a few AIPs from different countries and put the GEN and ENR sections next to each other, to see what I mean. And obviously the AD section, which is usually by far the biggest part of the AIP, is only needed if you fly to a specific airport. Still, it makes sense to compare the coverage of a few different airports with each other.
AIC - These documents contain "good advice" from the CAA that more often than not have no basis in any legal text. There are a few that the CAA requires that you are familiar with on the Air Law exam so you'd better be. But other than that the good advice can be found elsewhere as well. If you're really bored they actually form a good read on a rainy Sunday.
One you forgot - POH. Most aircraft (except the really old designs, or the non-certified/experimental ones) will have a POH and this POH is written to an international standard as well. Just as with the AIP, it's well worth getting hold of a few POHs of different aircraft types, and compare their contents. So you know where to look up information like landing distances and other performance figures in case it's ever needed - when preparing a flight for instance...
So, in short, yeah, I think it's very useful to become familiar with the structure of these documents, so you have an idea where to find things. But learning 684 pages of LASORS by heart is daft indeed - particularly since a lot of stuff in there is not related to PPL licensing at all.