Upvote:3
As a South Korean citizen, you can definitely enter either France or Italy and travel between the two for 90 days after the expiration of both your permit and your student visa so you certainly won't be staying illegally in France or Italy, be subject to removal or anything like that in September/October. For more details on this, see Getting a tourist Schengen visa after working visa (but the same rules apply to student visas).
Furthermore, there are no systematic passport checks between France and Italy so even if you are technically supposed to wait for the renewal for the permit in Italy, I don't think you would generally risk much by crossing that border during that time.
What I don't know is whether this whole visa/permit discrepancy makes any difference and whether the 90-day maximum stay limit runs from September 19 (it would be my guess) or from September 3. But that's moot as far as your trip to France and your return to Italy go.
What could however be important is applying for the renewal before the expiration of the current permit (so between August 31 and September 2?). Absolute worse case scenario if you fail to do would be that you would not be eligible for a permesso renewal application anymore and would be forced to go back to Korea to restart the process from scratch (new visa, etc.)
As long as you have the student visa, it could still be OK but I don't know what the rules around the renewal of a permesso di soggiorno exactly are and I am just speculating here. Questions about that are perhaps best suited for https://expatriates.stackexchange.com/