How can I check via software if my iPhone is unlocked?
So in the end I had to do the SIM card dance.

In fact, it seems what the carrier does is mark your phone as allowed for unlock.

Then, the first time you put a SIM card from another carrier, you must connect to iTunes, which will then perform a series of annoyances and alerts and whatnot, and only then your device will be truly unlocked. You need an internet connection for this process.

I know this is a terrible answer to my own question.