One, you have dips and sways in the floor that require a stiffer sub floor, but... Two, to add a stiffer sub flooring and not to remove the old makes a problem under doors, cabinets, and where the kitchen floor meets other adjoining room floors. With me??
Going over the asbestos is advisable, but how will you tackle the additional height?
If you have your existing floor dipping between floor joists, then you need to add 3/4 inch ply board. I would not use the wafer board or the particle board especially in the kitchen. When these wood products meet water they disolve.
Ok, 3/4 inch new sub floor and the new tile is 1/16 of an inch, then you have 13/16 inch new additional height.
Just something that you need to be thinking about.