I have an old CPXj that I've been using. I tried TinyXP on it and had the same results as you. I searched high and low and tried several drivers I found for the cardbus controller, but could never find one that worked properly.
I finally just gave up and went with a regular install of XP and disabled as many services from starting up as possible. It's certainly not as zippy as my newer laptops but it's usable.
I think there is a newer TinyXP version (6 ?) but I haven't tried it. If you find one that works please post the link info or the driver here.