How far out from the class start date do I need to be for reclass?

No, that's called a short fuse training seat. HRC uses those ones for mandatory reclasses. You have to choose class seats that are at least 60 days away. There's just not enough time for you to get the class seat, reenlist, come on assignment, have the assignment cap cycle, get a levy packet, turn in the levy packet, receive orders, turn in your orders to receive a DD 1610 and then receive the 1610.... Plus any clearing if you are single or going on an unaccompanied tour.

I locked in a class seat with an October 20th report date. I did this around mid-August, and I'm cutting it pretty close.

Really depends on how quickly your S3 can work it, and how quick you can clear. Would it be TDY and return to your unit? Or TDY en route to your new duty station?