If you install Steam version you actually need to start Steam?!...

You can play it by skipping Steam. Just install the launcher downloadable for the Free Trial from Square Enix and log-in as normal. The launchers are cross-compatible, the incompatibility seems to simply be artificial incompatibility of connecting the expansions with base game, nothing more.

Of course, that may mean you'll have to download the whole game from scratch.

EDIT: Guess I took to long. Anyway, I can confirm this works, at least. That's how I play having used the free trial first.