There are many things (mostly door stops and cabinets) that are anchored through the carpet into the subfloor with 3" screws in our Southwind. I would think the chair anchor, if it's like ours, would be enough to ensure that the tree wouldn't move and a second screw would prevent tip over.
You might be able to fashion a bar that would cross the base of the cat tree and screw the chair anchor with enough leverage to anchor it without any additional screws (the chair anchor is very substantial).