Is the Commander strictly fixed during deck construction?
It doesn't matter. Since you can change decks between games, you can change to a deck that's identical except for the choice of Commander.

Everyone talks about the Commander as if it's a property of the deck, including the Comprehensive Rules and the EDH site.

CR: In the Commander variant, each deck is led by a legendary creature designated as that deck’s commander.
EDH: Players choose a legendary creature as the &quot;Commander&quot; for their deck.

But that's not necessarily significant. It's simply the obvious and simple way to explain Commander. There's no explicit mention in the rules that the Commander is a property of the deck.
Does it matter?
In a casual setting, one may change decks between games as often as one desires, so you can obviously play two games with decks that vary only by your choice of Commander. The answer to your question is therefore irrelevant in a casual setting.
The answer is only potentially meaningful in a tournament setting, but I can't find anything suggestion that DCI hosts Commander tournaments. That means the the answer to your question is irrelevant as far as Wizard-sanctioned tournaments go.
That leaves non-sanctioned tournaments. Since they're non-sanctioned, there's no way to predict their deck registration policy. The answer is unanswerable for those in general.
The only tournament I was able to find is this league. But since they allow you to change deck between games, the answer to your question is irrelevant there too.

Given all of the above, my answer to your question is as follows:
If you enter a situation where you must reuse a deck for more than one game, consult with those with which you enter into this situation on this matter. Otherwise, it doesn't matter. Since you can change decks between games, you can change to a deck that's identical except for the choice of Commander.