Why not just stick it in the Room of Requirement or some hidden room that only the Headmaster can access (and use proper security)?

Because then (from Flamel's perspective) the Dark Lord would use the Crown of the Serpent and find that the Stone was not, in fact, at the end of the dungeon crawl, which would then nullify the purpose of the trap: to lure the Dark Lord there. (Presumably the Dark Lord would see no point in traipsing past all of those obstacles if the Stone wasn't there.)

Flamel would then insist upon the Stone being placed under Dumbledore's "best protections", meaning... huh. Is this series of obstacles really the best Dumbledore can do? For that matter, if Flamel taught Dumbledore so much of his powerful magic, presumably he/she knows much more powerful magic ("more" in terms of quantity, at least--not necessarily strength) that he/she hasn't seen fit to teach yet. Why not just protect the Stone himself?