My preferred notation method would simply be a small text note, perhaps as a symbol and footnote, indicating that the held note may be staggered if necessary. Specifically spelling out how to stagger it seems unnecessary, especially since simply holding out the note is also acceptable if possible.

It seems simple enough to write two parts for two different flute chairs. Simply add breath marks in alternating places. Then you can carry the note potentially for the entire piece.

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So, the note might be too long to play comfortably or at all. I was wondering whether I could give the illusion of this impossibly long note by requesting two players to collaborate. One will begin and fade out when necessary and the other would come in. A couple of such handovers should be sufficient.

In one of my ideas, I want a long sustained flute note to bridge two movements. The other instruments will stop on a chord but the flute will continue. The next movement will begin on another chord in which the flute's note serves a different role.

I don't think a flute is loud enough that you need to worry about one part fading in while another is fading out (if, as you describe, there are other instruments accompanying).

It might be an idea to keep in mind that the composer (especially one just starting out in the world) should write as practically for the chosen instruments as possible. The flute would not be my first choice of woodwind for holding a long and exposed note. The reed instruments are better at this for obvious reasons. Your long flute note might have to be so soft that it might not achieve your desired effect of maintaining audience attention. The dovetailing of two flutes would work but an oboe would be more noticeable and would achieve it effortlessly. If it's flute that you really want, then go for the two flutes.

In the event that you do manage to get a reading of your piece, it will likely be through a college or community band/orchestra, and I don't think it's reasonable to expect players at that level to circular breathe or be able to sustain a note for that long.

I agree with the other user that if this is just for fun and it's all going to stay in software, there's no reason to worry about this kind of practicality.

I sometimes compose for my own amusement but I have no delusions of creating anything that anyone else would like to hear.

As long as you are composing only for your amusement, the capabilities of MIDI expanders will be more important than that of actual flutists anyway.

If you are really interested in playability, then what you suggested is completely reasonable. Flutes in particular can sneak in and out very subtly. It might help to have at least two people playing at all times, since the difference between one and two players is noticeable.

Find a flutist who can do circular breathing. They are rarer than oboists with that skill, but it is getting increasingly demanded these days.