Of OP's four suggestions, only the first...
Their life's work
...is remotely credible, on the grounds of basic grammar.
The second fails because live isn't a noun. Only nouns can take the Saxon genitive 's.
The remaining two fail because they don't include any possessive marker at all. It's an integral part of the "base form" that isn't dependent on "plurality", so I see no justification for discarding it. Besides which, lifes in #4 isn't a valid plural except in special cases such as several still lifes (see Wikipedia).
In OP's specific context it's reasonable for work to be in the singular, because it applies to a single collaborative effort (the software company). But even where the reference is to multiple, independent efforts, people usually don't pluralise to works...
But seven artists stood out: Braque, Chardon, Duchamp, Hals, Peale, Sloan, and Trumbull produced 8% of their life's work in the 80s.
...dozens of scientists, spiritual leaders, and social activists make the telling of a sacred evolutionary story part of their life's work.
Having said that, I think pluralising is sometimes a perfectly justifiable choice...
[We] picked cartoonists whose life's works would be most familiar to the average citizen.
TL;DR: the collocation life's work is effectively an established single syntactic unit. You can modify the entire thing by adding a pluralising s (although in fact most people don't, even where semantically justified), but you can't tinker with the internal structure, so life's must remain unchanged.
Which is semantically sensible anyway, since each individual life is normally only dedicated to a single work, even if you're actually talking about the "works" of multiple "lives".