Just be wary of inducing too much grip/lift into the kart. Sometimes when the kart is flat and bogging on new tires, it’s because the added grip of the new tires is causing the kart to lift too harshly and it sets the inside rear tire down too quickly. On a kart with more power, this typically manifests as a hopping conditions, but in a 206 it may never generate the forces necessary to hop, and it might just sit flat.
If none of the above suggestions work, you may have to go the other way on your adjustments.
Your axle choice seems correct, as well as going to MXC wheels and dropping pressures. If the handling issue is happening right out of the gate on new rubber, it’s likely not going to be fixed with tire pressures or wheel choices, it’s a fundamental balance issue with the kart that needs to be adjusted.
If the above adjustments fail to improve it, try going with negative camber and widening the rear. The negative will help the kart roll better through the center of the corner and will reduce some of the initial harshness (if that’s the issue) on turn-in.