Apostrophe with Acronym

I think your question becomes much easier when you simplify it. With the Louisiana State Employees' Retirement System, you are dealing with a few factors that add confusion. One is that your acronym ends in an ess (since it part of an acronym, and thus represents the first letter of a word, I would treat it as I would any other letter); another is that you are using an acronym at all (this also shouldn't affect the need for an apostrophe); and the third is that you are referring to an organization that may or may not have "the" as part of its official title (though you didn't capitalize it, so I'm assuming that it doesn't). Replace LASERS with another company/system name that doesn't have any of these confusing attributes (none of which should affect the use of apostrophe): Interwoven.

Are you still uncertain about the apostrophes? If you are, we'll need examples with more context, such as full-sentence examples. The ones you gave are ambiguous, and in most cases have more than one correct answer, each of which means something different. For example, if the Web site is created using a technology called LASERS, you could say "I browsed Accenture's LASERS Web site." On the other hand, if you are talking about a Web site for a company called LASERS, you might say "I was looking at LASERS's Web site." I am pretty sure, however, that the possessive form of LASERS should always end with apostrophe followed by an ess.

if a regular word ends with ess the apostrophe should not be followed by ess as in "the lasers' wavelengths were invisible".
The issue might be whether an acronym is considered a regular word. At least some style guides prescribe it, tough I can't remember references. I'd consult the style guide of some well known newspapers, and, the style guide inside some of the larger dictionaries.

LASERS's is correct. The last 'S' in the acronym stands for "system". "Louisiana State Employees' Retirement System's" is the correct possessive. To not have the "s" after the apostrophe is to accept the LASERS acronym as an anacronym, which it's not.

Correct usages for your question:

LASERS's website

The LASERS Website (This is a proper noun, capitalize the first letters).

LASERS members (This is fine if not possessive; possessive would add an apostrophe after "members").

the LASERS's agencies (See above if possessive).

LASERS's agencies

LASERS's retiree billing, etc.